Patients with known central nervous system (CNS) involvement may be excluded; however, if CNS disease is cleared before the treatment with nivolumab, patients could be allowed if no permanent CNS damage Central nervous system (CNS) involvement; CNS status must be confirmed by lumbar puncture\r\n* Note: lumbar puncture can be performed at the time of diagnosis and does not need to be repeated unless there is a change in neurological status or it was performed more than 14 days prior to study entry Patient does not have any documented history of central nervous system (CNS) involvement by mantle cell lymphoma; this includes no evidence of parenchymal brain, spinal cord, or cerebrospinal fluid involvement; radiculopathy symptoms from nerve root compression by lymphoma do not constitute CNS involvement NOTE: A lumbar puncture is not required in order to be enrolled on study nor are lumbar punctures recommended at the time of diagnosis; if the diagnosis of APL is known or suspected, diagnostic lumbar punctures in patients with neurologic symptoms should be deferred until any coagulopathy is corrected; if central nervous system (CNS) disease is suspected or proven, a computed tomography (CT) or magnetic resonance imaging (MRI) should be considered to rule out the possibility of an associated chloroma; if CNS disease is documented, patients are still eligible and will receive protocol directed intrathecal treatments Patients must have a lumbar puncture to determine CNS involvement of ALL within 14 days prior to registration; patients with CNS3 are excluded from the trial; patients with CNS1 or CNS2 will be eligible, but will be monitored for CNS involvement; note that intrathecal methotrexate administered during the pre-study lumbar puncture may count as the first dose of intrathecal therapy required as part of the study Current clinical central nervous system (CNS) symptoms deemed by the investigator to be related to leukemic CNS involvement (no lumbar puncture required, clinical assessment per investigator’s judgment is sufficient) Patients must not have active central nervous system (CNS) involvement (by clinical evaluation); patients with previous documented history of CNS involvement of acute leukemia, or with clinical signs or symptoms consistent with CNS involvement of acute leukemia, must have a lumbar puncture which is negative for CNS involvement of acute leukemia; the lumbar puncture must be completed within 14 days prior to registration; patients with no previous history of documented CNS involvement and with no clinical signs or symptoms consistent with CNS involvement are not required to have completed a lumbar puncture before registration; note that treatment with intrathecal therapy is recommended during protocol treatment but CNS analysis during treatment is not required Subjects with known active or history of CNS involvement by malignancy Patients with known involvement of the CNS by malignancy will be included if there is no evidence of active bleeding or intratumoral hemorrhage on radiographic imaging Patients with asymptomatic CNS involvement are allowed. Symptomatic CNS involvement. Known CNS involvement at diagnosis Symptomatic central nervous system (CNS) involvement or lymphangitic metastasis. Stable or improving CNS disease that is not under active treatment after receipt of adequate therapy is allowed. Leptomeningeal disease as the only site of CNS involvement Active involvement of the central nervous system (CNS) by lymphoma; work-up for CNS involvement at diagnosis will be directed as per the treating physician and will depend on specific clinical circumstances (no brain imaging or lumbar puncture is required by this protocol) Patients with metabolic storage diseases who have severe central nervous system (CNS) involvement of disease, defined as intelligence quotient (IQ) score < 70 Subject has confirmed central nervous system (CNS) involvement by DLBCL. Subjects at risk for CNS involvement per Investigator assessment must receive prophylaxis. For subjects at risk, or with any neurological symptoms, testing for CNS involvement is required at Screening. Active CNS involvement Patients with known involvement of the CNS by malignancy will be excluded. Central Nervous System (CNS) involvement with multiple myeloma defined as csf positivity for plasma cells or a parenchymal CNS plasmacytoma Evidence of CNS involvement by NHL Known active central nervous system (CNS) involvement by malignancy. Subjects with prior CNS involvement with their lymphoma must have completed effective treatment of their CNS disease at least 3 months prior to enrollment with no evidence of disease clinically and at least stable findings on relevant CNS imaging Active CNS involvement of their malignancy. Active central nervous system (CNS) involvement by malignancy, except in cases of virus-associated malignancies with CNS involvement in which case the patient may benefit from the transplant to control the malignancy Active CNS involvement by malignant cells (? 2 months from the conditioning) CNS metastatic involvement Known central nervous system (CNS) involvement by lymphoma; patients at high risk for secondary CNS involvement but without neurologic symptoms suspected to be due to lymphoma are allowed to be enrolled and receive intrathecal chemotherapy including but not limited to methotrexate, cytarabine and glucocorticoids; patients who are enrolled and subsequently identified to have pathologic confirmation of CNS involvement by lymphoma may be continued on study at the discretion of the principal investigator Known clinically active central nervous system (CNS) involvement; prior evidence of CNS involvement successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment and there are no neurological signs of potential CNS involvement Known clinically active CNS involvement. Active central nervous system (CNS) involvement of leukemia (lumbar puncture not required to rule out CNS involvement if not suspected) Subjects with CNS involvement may be included on the study as long as they have not had any seizure activity in past 4 weeks CNS involvement with disease refractory to intrathecal chemotherapy Has known active central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement; subjects with prior CNS involvement by lymphoma must have remission of the CNS component of the lymphoma; these subjects must have a baseline magnetic resonance imaging (MRI) during screening without evidence of new or enlarging brain lesions and must not have any new or progressive neurologic symptoms Prior or active central nervous system (CNS) involvement by myeloma (eg leptomeningeal disease); screening for this, for example, by lumbar puncture, is only required if suspicious symptoms or radiographic findings are present Absence of active central nervous system (CNS) involvement; NOTE: pre-enrollment lumbar puncture not mandatory Active CNS disorder or seizure disorder or known CNS disease or neurologic symptomatology; in case of AML active CNS involvement as detected by lumbar puncture or neuro-imaging (only to be done if clinically indicated) Patients with known central nervous system (CNS) involvement may be excluded; however, if CNS disease is cleared before the treatment with nivolumab, patients could be allowed if no permanent CNS damage Known cytopathologically confirmed central nervous system (CNS) infiltration (in absence of suspicion of CNS involvement, lumbar puncture not required) CNS involvement with malignancy. Have prior or active central nervous system (CNS) involvement (e.g. leptomeningeal disease, parenchymal masses) with myeloma; screening for this (e.g. with lumbar puncture) is not required unless suspicious symptoms are present Patients diagnosed with primary central nervous system (CNS) or with CNS metastases/involvement, at any time during the disease course, are excluded from the study. Has known active central nervous system (CNS) involvement; subjects with previously treated CNS disease may participate provided they are stable (without evidence of CNS leukemia at the time of screening and any neurologic symptoms have returned to baseline) and are not using steroids for at least 7 days prior to trial treatment Patients with metastatic central nervous system (CNS) tumors are allowed provided that they are clinically stable for a period of 30 days prior to study entry and there is not a requirement for steroid (other than close to physiologic doses) or anti-convulsant therapy; patients with leptomeningeal involvement are excluded Subjects with known active central nervous system (CNS) involvement by malignancy; subjects with prior CNS disease that has been effectively treated will be eligible if treatment was completed at least 3 months prior to enrollment and there is no evidence of disease or stable abnormalities on repeat imaging Subjects with known active CNS involvement by malignancy; subjects with prior CNS disease that has been effectively treated will be eligible if treatment was completed at least 3 months prior to enrollment and there is no evidence of disease or stable abnormalities on repeat imaging For lymphodepletion chemotherapy, JCAR014 and durvalumab: Subjects with known active CNS involvement by malignancy; subjects with prior CNS disease that has been effectively treated will be eligible if treatment was completed at least 3 months prior to enrollment and there is no evidence of disease or stable abnormalities on repeat imaging Clinical evidence of (parenchymal or meningeal) central nervous system (CNS) involvement or metastasis; in subjects suspected of having CNS disease, a magnetic resonance imaging (MRI) scan of the brain and lumbar puncture should be done to confirm Active central nervous system (CNS) lymphoma within two weeks of registration; patients with a history of CNS involvement must have adequate treatment as defined by at least two negative spinal fluid assessments separated by at least one week; (otherwise lumbar puncture [LP] is not required if no clinical suspicion or evidence of CNS involvement); patients who have received cranial radiation therapy must still be eligible to receive total lymphoid irradiation to 7 Gy Patients with active central nervous system (CNS) involvement with malignancy; patients with prior CNS disease that has been effectively treated will be eligible providing treatment was > 4 weeks before enrollment Subjects with signs or symptoms indicative of CNS involvement; a CNS evaluation should be performed as clinically appropriate to rule our CNS involvement Subjects must not have prior or active central nervous system (CNS) involvement (e.g. leptomeningeal disease, parenchymal masses) with myeloma; screening for this (e.g. with lumbar puncture) is not required unless suspicious symptoms are present History of CNS involvement by MM Current CNS involvement by disease Active central nervous system (CNS) involvement by malignancy; Note: Patients with history of CNS disease that has been effectively treated will be eligible provided that treatment was > 4 weeks before enrollment Patients with active central nervous system (CNS) involvement by malignancy; patients with prior CNS disease that has been effectively treated will be eligible providing treatment was > 4 weeks before enrollment Skin involvement. Participants with known central nervous system (CNS) involvement with leukemia or who are receiving intrathecal chemotherapy that is either prophylactic or therapeutic; history of CNS involvement that has been completely treated (no longer receiving intrathecal chemotherapy) will be allowed Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning (day -6) Evidence for active CNS involvement by leukaemia Patients diagnosed with primary central nervous system (CNS) or with CNS metastases or involvement, at any time during the disease course, are excluded from the study. Known central nervous system (CNS) lymphoma or leukemia; subjects with symptoms of CNS disease must have a negative computed tomography (CT) scan or negative diagnostic lumbar puncture prior to first dose 210 Evidence of CNS involvement by NHL. Subjects with central nervous system (CNS)-only involvement by malignancy (note: subjects with secondary CNS involvement are allowed on study) Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy; diagnostic lumbar puncture is to be performed Patient has a known or suspected primary central nervous system (CNS) tumor or CNS tumor involvement or active leptomeningeal disease. Known cytopathologically confirmed CNS infiltration (in absence of suspicion of CNS involvement, lumbar puncture not required) Patients with suspected central nervous system (CNS) involvement should undergo lumbar puncture; those with documented CNS involvement will be excluded Patients with untreated malignant involvement of the central nervous system (CNS) should be excluded from this clinical trial; head imaging will be necessary to document absence of CNS involvement in patients with colon/rectal cancer and soft tissue sarcomas; patients with hematologic malignancies who have undergone treatment for malignant involvement of the CNS must have no evidence of residual disease by imaging or cerebrospinal fluid (CSF) sampling prior to study enrollment Central nervous system involvement; a lumbar puncture does not need to be performed unless there is clinical suspicion of leptomeningeal disease With central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy, cranial irradiation or both prior to initiating conditioning (day -6) Patients with metabolic storage diseases who have severe central nervous system (CNS) involvement of disease, defined as intelligence quotient (IQ) score < 70 History of known leptomeningeal involvement (lumbar puncture not required) Patients must not have clinical evidence of central nervous system (CNS) involvement by lymphoma \r\n* Any laboratory or radiographic tests performed to assess CNS involvement must be negative and must be performed within 42 days prior to registration Patients with active CNS leukemia involvement defined as CNS-3 by CSF findings only are eligible but will have their CTL019 infusion delayed until CNS disease is reduced to CNS-1 or CNS-2 by CSF findings. Patients with other forms of active CNS-3 leukemic involvement such as CNS parenchymal or ocular disease, cranial nerve involvement or significant leptomeningeal disease are not eligible. However, such patients with other forms of CNS-3 leukemic involvement (non-CSF involvement) are eligible if there is documented evidence of disease stabilization for at least 3 months prior to CTL019 infusion. Patients must have no acute/ongoing neurologic toxicity > Grade 1 with the exception of a history of controlled seizures or fixed neurologic deficits that have been stable/improving over the past 3 months. Active CNS involvement by malignancy, defined by CNS-3 per NCCN guidelines. Active signs or symptoms of CNS involvement by malignancy Patients with primary CNS tumor or CNS tumor involvement Evidence of CNS involvement with DLBCL CNS lymphoma involvement A primary lymphoma of the central nervous system (CNS) or known lymphomatous involvement of the CNS. CNS imaging and cerebrospinal fluid sampling are not mandatory in the absence of a clinical suspicion of lymphomatous involvement of the CNS. Patients must not have active involvement of the central nervous system (CNS) with MM (by clinical evaluation); patients with documentation of, or clinical signs or symptoms consistent with, CNS involvement of MM must have a lumbar puncture that is negative for CNS involvement of MM; the lumbar puncture must be completed within 14 days prior to registration; patients with no previous history of documented CNS involvement and with no clinical signs or symptoms consistent with CNS involvement are not required to have completed a lumbar puncture prior to registration; note that monitoring of CNS involvement and treatment with intrathecal therapy is recommended during protocol treatment History of, or known central nervous system (CNS) involvement caused by the underlying B-cell malignancy or prior history of National Cancer Institute common terminology criteria for adverse events (NCI CTCAE) Grade greater than or equal to >= 3 drug-related CNS toxicity. Participants with signs or symptoms of CNS involvement should have a computed tomography (CT) or magnetic resonance imaging (MRI) Patients with active central nervous system (CNS) involvement by malignancy; patients with prior CNS disease that has been effectively treated will be eligible providing treatment was > 4 weeks before enrollment Participants with active central nervous system (CNS) involvement are excluded; patients with suspected CNS disease should be worked up appropriately prior to enrollment Primary CNS lymphoma or evidence of CNS involvement by lymphoma Known central nervous system lymphoma or leukemia; subjects with symptoms of CNS disease must have a negative CT scan or negative diagnostic lumbar puncture prior to randomization Primary central nervous system (CNS) lymphoma or known or suspected CNS involvement by non-primary CNS NHL Subjects with known active central nervous system (CNS) involvement by malignancy. Those with prior CNS disease that has been effectively treated will be eligible if treatment was completed at least 3 months prior to enrollment with no evidence of symptomatic disease and stable abnormalities on repeat imaging. Patients with a history of central nervous system (CNS) leukemia are eligible if they are not symptomatic from current CNS involvement\r\n* If there is CNS involvement that is known prior to enrollment or identified subsequently, it will be treated accordingly Patients with active central nervous system (CNS) involvement with malignancy; patients with prior CNS disease that has been effectively treated will be eligible providing treatment was > 4 weeks before enrollment Current clinical central nervous system (CNS) symptoms deemed by the investigator to be related to leukaemic CNS involvement (no lumbar puncture required, clinical assessment per investigator´s judgement is sufficient). Absence of CNS involvement by leukemia. Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning; diagnostic lumbar puncture is to be performed per protocol Untreated central nervous system (CNS) involvement; (treated CNS involvement is permitted only if the patient is not currently on steroid therapy or has remained on a stable, unchanged dose of steroid for >= 2 weeks) Known cytopathologically confirmed central nervous system (CNS) infiltration (in absence of suspicion of CNS involvement, lumbar puncture not required) Known active CNS involvement Known CNS involvement of indolent lymphoma Known or clinically suspected CNS involvement Symptomatic CNS involvement (other than signs and symptoms caused by leptomeningeal disease) Active central nervous system (CNS) involvement by malignancy (defined as CNS-3 per National Comprehensive Cancer Network [NCCN] guidelines) CNS or leptomeningeal involvement of lymphoma Known Central Nervous System (CNS) lymphoma; subjects with symptoms of CNS disease must have a negative CT scan and negative diagnostic lumbar puncture Patients with tumor involvement of the Central Nervous System (CNS). SCLC patients with previously treated CNS lesions must have stable CNS disease for at least 4 weeks Known active and uncontrolled central nervous system (CNS) involvement of leukemia (a lumbar puncture does not need to be performed as a part of screening) Untreated central nervous system (CNS) metastatic disease as defined by:\r\n* Solid tumors: history of untreated metastatic CNS tumor involvement; extradural masses which have not invaded the brain parenchyma or parameningeal tumors without evidence for leptomeningeal spread will not render the patient ineligible; patients with previous CNS tumor involvement are eligible IF the CNS tumor(s) has been treated and has been stable or resolving for at least 4 weeks; and if the patient does not currently require steroids Patients with uncontrolled CNS tumor metastatic involvement Known central nervous system (CNS) leukemia by spinal fluid cytology, flow cytometry or imaging; a lumbar puncture is not required unless CNS involvement is clinically suspected; patients with signs or symptoms of leukemic meningitis must have a negative lumbar puncture within 2 weeks of study enrollment Patient with CNS involvement unless they are at least 4 weeks from prior therapy completion. Patients with prior central nervous system (CNS) involvement are eligible provided that it has been treated and is in remission; CNS therapy (chemotherapy or radiation) should continue as medically indicated during the protocol Known, clinically suspected, or history of CNS tumor involvement. Patients with known central nervous system (CNS) leukemia by spinal fluid cytology, flow cytometry or imaging; a lumbar puncture is not required unless CNS involvement is clinically suspected; patients with signs or symptoms of leukemic meningitis or a history of leukemic meningitis must have a negative lumbar puncture within 2 weeks of study enrollment Patients has primary CNS tumor or CNS tumor involvement Symptomatic central nervous system (CNS) metastases or carcinomatous meningitis. Asymptomatic patients must be clinically stable with regard to their CNS/ meningeal metastatic involvement, have completed previous therapy (including radiation and/ or surgery) at least 4 weeks prior to study drug administration, are not receiving steroid therapy or taper, and are not receiving anti-convulsive medication for any CNS involvement Patients with known central nervous system (CNS) or leptomeningeal metastases not controlled by prior surgery or radiotherapy, or patients with symptoms suggesting CNS involvement for which treatment is required. Evidence of current CNS involvement. Patients with known CNS (or leptomeningeal) metastases not controlled by prior surgery or radiotherapy, or patients with symptoms suggesting CNS involvement for which treatment is required. Clinical evidence of CNS involvement by lymphoma Primary CNS tumor or CNS tumor involvement Recipient must have adequate neurologic function as defined by NO evidence of a severe central or peripheral neurologic abnormality. Patients with a history of previous CNS tumor involvement are eligible provided they are without symptoms or signs and the CNS is now free of disease on lumbar puncture and CT scan of the brain. Presence of CNS involvement of leukemia. Patients with a history of CNS involvement may be considered after discussion with the Medical Monitor Presence of clinically evident active CNS metastases, including leptomeningeal involvement, requiring steroid or radiation therapy Presence of clinically evident active CNS metastases, including leptomeningeal involvement, requiring steroid or radiation therapy The patient was diagnosed with leukemic Central Nervous System (CNS) disease (e.g. before chemotherapy) or presents neurological symptoms at baseline suggestive of a CNS involvement. Prior central nervous system (CNS) involvement by tumor is permissible if previously treated and clinically stable for two weeks after completion of treatment. Patients with a history of CNS (central nervous system) leukemia are eligible if they are not symptomatic from current CNS involvement; if there is CNS involvement that is known prior to enrollment or identified subsequently, it will be treated accordingly with intrathecal chemotherapy per the treating physician History of known CNS leukemia, including cerebrospinal fluid positive for AML blasts; lumbar puncture is recommended for subjects with symptoms of CNS leukemia to rule out extramedullary CNS involvement; Active CNS leukemia. Patients with symptoms of CNS involvement, particularly those with M4 or M5 subtypes, should undergo lumbar puncture prior to treatment on study to exclude CNS disease. Symptoms include cranial neuropathies, other neurologic deficits, and headache.