Eastern Cooperative Oncology Group (ECOG) performance status of ? 2 for Part 1 and ? 1 for Part 2. Adequate organ function. Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 within 2 weeks prior to screening Patients must have Eastern Cooperative Oncology Group (ECOG) performance status 0-1\r\n* NOTE: Participants with impaired decision-making capacity (IDMC) should not be allowed to participate in this study due to its complexity Eastern Cooperative Oncology Group (ECOG) performance status of =< 1 within 90 days prior to step 1 registration Eastern Cooperative Oncology Group (ECOG) performance status 0-1 60 days prior to registration Eastern Cooperative Oncology Group (ECOG) status 0-2 within 1 month prior to registration Eastern Cooperative Oncology Group (ECOG) performance status of 0-3 within 7 days of enrollment Eastern Cooperative Oncology Group (ECOG) performance status of zero to two unless decline is due to bony disease ELIGIBILITY CRITERIA - PHASE I (ARMS A, B, C): Eastern Cooperative Oncology Group (ECOG) performance status 0-2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days prior to registration Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1 within 10 days prior to the start of study treatment. Patients must be ambulatory with good performance status (Eastern Cooperative Oncology Group [ECOG] 0 or 1) INCLUSION CRITERIA FOR SECOND-LINE THERAPY: Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 Eastern Cooperative Oncology Group (ECOG) performance status must be ?2 (Appendix A). SAFETY RUN-IN: Eastern Cooperative Oncology Group (ECOG) performance status 0-2 Eastern cooperative oncology group (ECOG) performed status of 0, 1 or 2; 0 or 1 (cohort E only) Eastern Collaborative Oncology Group (ECOG) Performance Status of 0-1 (Appendix 1). Eastern Cooperative Oncology Group (ECOG) performance status =< 3 or Lansky performance status of >= 30 Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1 within 10 days of randomization. Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 10 days of study start ENROLLMENT TO THE DOSE ESCALATION, EXPANSION AND PART II: Eastern Cooperative Oncology Group (ECOG) performance status =< 2 Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 within 28 days prior to study registration. Eastern Cooperative Oncology Group (ECOG) performance status 0 – 2 (asymptomatic to symptomatic but capable of self-care) within 45 days prior to randomization Eastern Cooperative Oncology Group performance status of less than or equal to ( 2 months Eastern Cooperative Oncology Group (ECOG) performance status of =< 2 and fit for induction therapy in the opinion of the treating physician Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 or better History/physical examination, including breast exam and documentation of weight and Eastern Cooperative Oncology Group (ECOG) performance status Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, 2, or 3, if the performance status of 3 is due to skin disease involvement Eastern Cooperative Oncology Group (ECOG) performance status scores of 0, 1, or 2 Eastern Cooperative Oncology Group (ECOG) performance status of =< 1 within 14 days prior to being registered for protocol therapy The subject must have a performance status of 0-1 as determined by criteria set forward by the Eastern Cooperative Oncology Group Eastern Co-operative Oncology Group (ECOG) performance status of ? 2. Patients must have a performance status of 0 to 2 according to the Eastern Cooperative Oncology Group (ECOG) criteria Current Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 through screening to first treatment on study Eastern Cooperative Oncology Group (ECOG) performance status of 2 or better CAPMATINIB INCLUSION CRITERIA: Eastern Cooperative Oncology Group (ECOG) performance status 0-1 Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2; performance status of 3 permissible if related to disease Eastern Cooperative Oncology Group (ECOG) performance status 0,1 (Appendix D) Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 - 1 within 30 days of signing informed consent PHASE I STUDY ELIGIBILITY CRITERIA:\r\nEastern Cooperative Oncology Group (ECOG) performance status =< 2 Eastern Cooperative Oncology Group (ECOG) performance status of zero, one, or two Eastern Cooperative Oncology Group (ECOG) performance status 0–2 (3 if it is directly disease related and is expected to get better if the acute lymphoblastic leukemia/lymphoma [ALL] is under control) Eastern Cooperative Oncology Group (ECOG) performance status must be level 0 or 1 within 60 days prior to registration Adequate performance status (Eastern Cooperative Oncology Group [ECOG] < 3) and research participant does not require mechanical ventilation Eastern Clinical Oncology Group performance status 0 or 1 Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0-2 within 30 days of signing informed consent (Turnstile I) Eastern Clinical Oncology Group performance status 0 or 1 Eastern Cooperative Oncology Group (ECOG)-American College of Radiology Imaging Network (ACRIN) performance status between 0-2 Relapsed/refractory MCL: Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less Newly diagnosed MCL: Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less Patient eligible for standard induction chemotherapy based on Eastern Cooperative Oncology Group (ECOG) performance status (PS) and vital organ function at the discretion of the treating physician Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0-2 within 30 days of signing informed consent (Turnstile I) Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 within 7 days prior to registration Eastern Cooperative Oncology Group (ECOG) performance status 2 or better Patients must be ambulatory with an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 Eastern Cooperative Oncology Group (ECOG) status 0-1 documented within 90 days of registration Participants will not be excluded based on performance status; for patients with Eastern Cooperative Oncology Group (ECOG) performance status >= to 3 the dosing schedule will be discussed with study chairman Patients > 70 may be considered if performance status > 80% or Eastern Cooperative Oncology Group (ECOG) =< 1 and comorbidity score < 3; these patients must be discussed with the principal investigator (PI), Rachel Salit prior to enrollment Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0-2 (Turnstile I) Easter Cooperative Oncology Group (ECOG) Performance Status ? 1 Eastern Cooperative Oncology Group (ECOG) performance status grade less than or equal to (<=) 2 Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 within 14 days prior to registration for protocol therapy Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2 at time of enrollment. Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 or 2 within 28 days prior to registration ORAL CAVITY SQUAMOUS CELL CARCINOMA COHORT: Eastern Cooperative Oncology Group (ECOG) performance status < 2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 7 days prior to the first dose of study treatment but before randomization. Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2, and estimated survival of at least 3 months Eastern Cooperative Oncology Group (ECOG) performance status of =< 3 and fit for induction therapy in the opinion of the treating physician REGISTRATION TO TREATMENT (STEP 1): Patients have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 Patients must have Eastern Cooperative Oncology Group (ECOG) performance status 0-2 and expected survival > 3 months from study enrollment Eastern Cooperative Oncology Group (ECOG) performance status being 0-1 within 1 week of planned start of therapy Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 within 28 days prior to registration FOR TISSUE COLLECTION TO ESTABLISH PDX (PART 1): Eastern Cooperative Oncology Group (ECOG) performance status of 0-2. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 28 days prior to registration for protocol therapy The subject must have a performance status of 0-2 as determined by criteria set forward by the Eastern Cooperative Oncology Group DOSE ESCALATION COHORT: Eastern Cooperative Oncology Group (ECOG) performance status 0-2 Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 within 30 days prior to registration. Eastern Cooperative Oncology Group (ECOG) performance status 0-2 at time of study entry PSTAT3 SCREENING: Eastern Cooperative Oncology Group (ECOG) performance status 0-2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days before registration for protocol therapy Performance status of 2 or better as per Eastern Cooperative Oncology Group (ECOG) criteria Easter Cooperative Oncology Group (ECOG) performance status 0-2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at initial enrollment, as assessed by clinician or investigator Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 - 1; ECOG performance status of 0-1 will be inferred if the patient’s level of energy is >= 50% of baseline Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 - 1; ECOG performance status of 0-1 will be inferred if the patient's level of energy is >= 50% of baseline Performance Status: Eastern Cooperative Oncology Group (ECOG) 0-1 unless mobility is limited from chronic physical handicap Eastern Cooperation Oncology Group (ECOG) performance status 0-2 Patients must have Eastern Cooperative Oncology Group (ECOG)-American College of Radiology Imaging Network (ACRIN) performance status of 0-2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 10 days prior to study start Eastern Cooperative Oncology Group (ECOG) performance less than or equal to 2 Eastern Cooperative Oncology Group (ECOG) performance status 0, 1 or 2 at the time of informed consent The participant has an Eastern Cooperative Oncology Group performance status of 0 or Eastern Cooperative Oncology Group (ECOG) performance status of 0-2; patients with ECOG performance status of 3 will be eligible if the lower performance status is deemed by the investigator to be due entirely to accelerated or blastic phase MPN and not due to another comorbidity Patients who have received one prior regimen must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2; patients who have received two or three prior regimens must have an ECOG performance status of 0 or 1 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 10 days prior to study start Performance status of 2 or better as per Eastern Cooperative Oncology Group (ECOG) criteria AMKL PATIENTS: Patients must have an Eastern Cooperative Oncology Group (ECOG) status 0-2 Eastern Cooperative Oncology Group (ECOG) Performance status less than or equal to (= 16 or Lansky play >= 80% for ages =< 15 World Health Organisation (WHO) or Eastern Cooperative Oncology Group (ECOG) Performance status 0-2. Easter Cooperative Oncology Group (ECOG) Performance Status of 0-2 The Eastern Cooperative Oncology Group (ECOG)/World Health Organization (WHO) performance status of 2-3.or patients with chronic lymphocytic leukemia requiring treatment age >65 years at the time of signing informed consent and ECOG/WHO performance status of 0-1 and a CIRS or Charlson co-morbidity score of 2 or higher. Eastern Cooperative Oncology Group (ECOG) performance status of =< 2 or greater, based on treating physician's discretion Have a performance status less than or equal to 1 for Dose Escalation (Part A) and less than or equal to 2 for Dose Confirmation (Parts B, C, D, E, F and G) on the Eastern Cooperative Oncology Group (ECOG) scale Clinical performance status of Eastern Cooperative Oncology Group (ECOG) less than or equal to 1 Has Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 performed within 10 days of treatment initiation. Have Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 (see Appendix 14.3) Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 within 14 days prior to registration Easter Cooperative Oncology Group (ECOG) performance status of 0,1,or 2 Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 10 days prior to registration for protocol therapy Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 at the time of screening and enrollment Eastern Cooperative Oncology Group (ECOG) Performance Status of 0 or 1 within 28 days prior to registration. Subject has Eastern Cooperative Oncology Group (ECOG) performance status of less than or equal to 2 at screening. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days before enrollment. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days before enrollment. Eastern Cooperative Oncology Growth (ECOG) performance status =< 2 Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days prior to the initiation of study treatment Eastern Cooperative Oncology Group (ECOG) performance status measure will be used; (ECOG performance status less than or equal to 3) Eastern Cooperative Oncology Group (ECOG) performance status 0 to 1 within 14 days of study entry Eastern Cooperative Oncology Group performance status less than or equal to 2. Patients must have a Eastern Cooperative Oncology Group status of 50% of the day) Eastern Cooperative Oncology Group (ECOG) performance status from 0 (asymptomatic) to 2 (symptomatic and in bed < 50% of day) Patient is spending more than 50% of time out of bed on a daily basis, as measured by an Eastern Cooperative Oncology Group (ECOG) performance status rating of level 0, 1, or 2 Eastern Cooperative Oncology Group (ECOG) performance status 0-1 during dose finding, 0-2 during MTD expansion Has Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 performed within 10 days prior to receiving the first dose of study therapy Eastern Cooperative Oncology Group (ECOG) performance status from 0 (asymptomatic) to 2 (symptomatic and in bed < 50% of the day) Eastern Cooperative Oncology Group (ECOG) performance status 0, 1 or 2 at time of initial screening Patients are eligible regardless of Eastern Cooperative Oncology Group (ECOG) performance status There will be no discrimination based on gender, race, creed, or ethnic background; Eastern Cooperative Oncology Group (ECOG)/Karnofsky performance scale (KPS) will not be employed Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1 within 28 days prior to initiation of treatment Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 (0=participant has either normal activity, 1= participant has some symptoms but is nearly full ambulatory) Eastern Cooperative Oncology Group (ECOG) rating of >= 3 or Karnofsky performance status (KPS) < 60 as rated by the oncology provider Eastern Cooperative Oncology Group (ECOG) performance status from 0 (asymptomatic) to 2 (symptomatic and in bed < 50% of the day) SCREENING PHASE: Eastern Cooperative Oncology Group performance status 0-2 INTERVENTION PHASE: Eastern Cooperative Oncology Group performance status 0-2 Eastern Cooperative Oncology Group (ECOG) performance status 0-1 (within 90 days of enrollment) Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (at the time the subject signs their first informed consent form); Easter Cooperative Oncology Group (ECOG) performance status 0 or 1 Eastern Cooperative Oncology Group (ECOG) ranging from 0 (asymptomatic) to 2 (symptomatic and in bed > 50% of the day) Eastern Cooperative Oncology Group performance status of 0 or 1 as assessed within 21 days prior to randomization. Patients with an Eastern Cooperative Oncology performance status of 0 or 1 Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 at Screening or by the day chemotherapy is initiated Eastern Cooperative Oncology Group (ECOG) performance status 2 or better Eastern Cooperative Oncology Group (COG) performance status of 0-2 Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (fully active or restricted only in physically strenuous activity) Eastern Cooperative Oncology Group (ECOG) performance status >2 (See Appendix G) Eastern Cooperative Oncology Group (ECOG) performance status of =< 2 within 28 days prior to registration Eastern Cooperative Oncology Group (ECOG) performance status of 0 - 1 within 28 days of registration The Eastern Cooperative Oncology Group (ECOG) performance status of 1 or better The patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 or, based on judgment of the treating medical oncologist, can tolerate imaging and at least 6 months of ET Eastern Cooperative Oncology Group (ECOG) performance status of =< 1 within 14 days prior to being registered for protocol Eastern Cooperative Oncology Group (ECOG) or Karnofsky performance status will not be employed, but patients will need to be sufficiently healthy to tolerate all study procedures AIM 3: Participants must be Eastern Cooperative Oncology Group (ECOG) status 0-3 Has Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1, with no deterioration over the previous 2 weeks Easter Cooperative Oncology Group (ECOG) performance status 0 or 1 Non-metastatic GI cancers are eligible only if the Eastern Cooperative Oncology Group (ECOG) performance status is 2, 3, or 4 at the time of consent (although patients with locally advanced [stage III] and inoperable pancreatic cancers are eligible regardless of their performance status) Women and men ?18 years of age with Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1, as assessed within 10 days prior to the start of study drug.