Patients with testing that did not show dMMR (loss of MMR protein) are not eligible to participate; patients whose tumors show MSI-H by polymerase chain reaction (PCR)-based assay are not eligible to participate unless they also have MMR testing by IHC and are found to have dMMR (i.e. loss of one or more MMR proteins) Tumor O()-methylguanine-DNA-methyltransferase (MGMT) methylation status must be available; results of routinely used methods for MGMT methylation testing (e.g. mutagenically separated polymerase chain reaction, MSPCR, or quantitative polymerase chain reaction [PCR]) are acceptable Subject has confirmed sufficient expression of NY-ESO- and/or LAGE-a by reverse transcription polymerase chain reaction (RT-PCR) as determined by a central laboratory contracted by the Sponsor (this determination will also be made under a pre-enrollment screening ICF). Participants with CD counts > /microL are eligible for this study if their viral load is < copies/mL by reverse transcription polymerase chain reaction (RT-PCR) since majority of the participants have received aggressive chemotherapy that can potentially decrease the CD counts despite the ART therapy; timeline: within weeks prior to start of trial Metastatic colorectal cancer (mCRC) categorized as microsatellite stable (MSS) by polymerase chain reaction (PCR) per local assay at any time prior to Screening or by the central laboratory. Microsatellite stable disease determined by IHC and/or polymerase chain reaction (PCR). Have confirmed human papillomavirus-associated lesions based on in-situ hybridization testing and/or polymerase chain reaction which may be performed on a newly obtained biopsy or archived sample Dose expansion:\r\n* HPV-associated locally advanced or metastatic platinum-resistant solid tumor malignancy; HPV positivity defined by positive p immunohistochemistry, polymerase chain reaction, or in-situ hybridization assessment of archival tissue (primary or metastatic) in a Clinical Laboratory Improvement Act (CLIA)-certified laboratory; availability of pathology report from CLIA-certified lab demonstrating positive HPV status by p IHC, polymerase chain reaction, or in situ hybridization qualifies for eligibility determination; analysis of fresh tumor tissue is permitted in cases where archival tissue is not available\r\n* Platinum resistance defined as prior progression (radiographic or clinical) either during or within months following completion of platinum-based chemotherapy\r\n* Platinum-based therapy as most recent systemic therapy prior to enrollment allowed but not required Patients must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis Tumor expression of NY-ESO- or LAGE- by immunohistochemistry (IHC) and/or reverse transcriptase polymerase chain reaction (RTPCR) p or HPV negative oropharyngeal squamous cell carcinoma (OPSCC) as determined by immunohistochemistry (IHC) and polymerase chain reaction (PCR), respectively Microsatellite instability (MSI) phenotype of archival tissue biopsy determined by treating institution by polymerase chain reaction (PCR) and immunohistochemistry (IHC) assay Known microsatellite stable (MSS) status by either immunohistochemistry (IHC) or polymerase chain reaction (PCR). Known or evaluable BRAF and KRAS status. Asymptomatic adenovirus viremia defined as no symptoms of adenovirus disease and EITHER two positive and quantifiable quantitative polymerase chain reaction (qPCR) tests taken one week apart or one single measurement with >= copies. Serologic status and/or polymerase chain reaction (PCR) testing reflecting active hepatitis B or C infection Active Hepatitis B (by surface antigen expression or polymerase chain reaction) or C (by polymerase chain reaction) infection or on hepatitis-related antiviral therapy within months of first dose of study drug. Tumor O--methylguanine-deoxyribonucleic acid (DNA) methyltransferase (MGMT) methylation status must be available; results of routinely used methods for MGMT methylation testing (e.g. mutagenically separated polymerase chain reaction [MSPCR] or quantitative polymerase chain reaction [PCR]) are acceptable EGFRvIII, the target antigen, must be identified on tumor tissue by immunohistochemistry (IHC) or polymerase chain reaction (PCR), i.e. EGFRvIII positive via pathology report Vulvar HSIL must be HPV-+ by a polymerase chain reaction (PCR), ribonucleic acid (RNA), or in situ hybridization test from a CLIA certified laboratory for subjects with MCL (confirmed with cyclin D expression or evidence of t(;) by cytogenetics, fluorescent in situ hybridization (FISH) or polymerase chain reaction (PCR): relapsed or refractory disease after at least prior regimen with chemo-immunotherapy (prior auto-HSCT is allowable) Must have MS/MMR result available at time of registration; MS/MMR status is to be determined per local practice (i.e. immunohistochemistry [IHC], polymerase chain reaction [PCR], or other methods) Patients should have microsatellite stable (MSS) tumor by polymerase chain reaction (PCR) assay or mismatch repair protein proficient (MMRP) tumor by immunohistochemistry as confirmed by the presence of MLH, MSH, MSH, and PMS; the diagnosis of colorectal cancer should be confirmed by pathology either on the primary tumor or from a prior biopsy of a metastatic disease site Participants must have histologically or cytologically confirmed squamous cell carcinoma of the oropharynx, hypopharynx, supraglottic larynx, nasal cavity, unknown primary, or nasopharynx that is p and HPV positive; tissue or cytology from the primary site or lymph node must be available for biomarker studies and for polymerase chain reaction (PCR) testing; immunohistochemistry (IHC) must be performed in a lab verified by the central laboratory or the slides must be available for review by the central laboratory (Zhang, MSSM) and PCR must be done in the central laboratory prior to radiotherapy; HPV PCR must be performed and results available for reduced dose therapy after induction.\r\n* Patients who are on the Quarterback Trial when Quarterback is activated and who have been randomized to radiotherapy arm will be asked to transfer to this trial and receive the Quarterback defined radiotherapy Histologically or cytologically proven HPVOC or cervical cancer or anal cancer, based on the presence of HPV type, detected by immunohistochemistry with P staining followed by polymerase chain reaction (PCR) of tumor tissue from the primary or metastatic lesions Measurable metastatic or refractory/recurrent HPV-+ cancer (determined by in situ hybridization [ISH] or a polymerase chain reaction [PCR]-based test) Documentation of HPV tested by polymerase chain reaction (PCR) Measurable metastatic cancer that expresses NY ESO- as assessed by one of the following methods: reverse transcriptase-polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO Expression of one () or more of the following TAPAs: SP, AKAP, Ropporin, PTTG and Span-xb, by either reverse transcriptase polymerase chain reaction (RT-PCR) and/or immunocytochemistry, Western blotting or ELISA, in neoplastic cells and/or blood. The presence of the target antigen, EGFRvIII, must be identified on tumor tissue by immunohistochemistry (IHC) or polymerase chain reaction (PCR) Detectable circulating tumor cells (CTCs) with detectable AR?V splice?variant by reverse transcriptase (RT)?polymerase chain reaction (PCR) Patients with microscopic hematuria OR biopsy proven BK nephritis and urine or blood polymerase chain reaction (PCR) positive for BK virus and/or JC viral encephalitis Metastatic or locally advanced refractory/recurrent cancer that expresses MAGE-A as assessed by one of the following methods: reverse transcriptase (RT)-polymerase chain reaction (PCR) on tumor tissue defined as , copies of MAGE-A per ^ glyceraldehyde--phosphate dehydrogenase (GAPDH) copies, or by immunohistochemistry of resected tissue defined as % or greater of tumor cells being -+ for MAGE-A, or serum antibody reactive with MAGE-A; metastatic cancer diagnosis will be confirmed by the Laboratory of Pathology at the National Cancer Institute (NCI) MRD will be defined in this protocol by presence of malignant cells at .% or more by flow cytometry or polymerase chain reaction (PCR) analysis at the completion of initial remission induction therapy Metastatic or locally advanced refractory/recurrent cancer that expresses MAGE-A as assessed by one of the following methods: reverse transcriptase (RT)-polymerase chain reaction (PCR) on tumor tissue defined as , copies of MAGE-A per ^ glyceraldehyde -phosphate dehydrogenase (GAPDH) copies, or by immunohistochemistry of resected tissue defined as % or greater of tumor cells being -+ for MAGE-A, or serum antibody reactive with MAGE-A; metastatic cancer diagnosis will be confirmed by the Laboratory of Pathology at the National Cancer Institute (NCI) The patient's tumor is HPV positive by polymerase chain reaction (PCR) or in situ hybridization (ISH) assay of tumor biopsy Measurable metastatic cancer or locally advanced refractory/recurrent malignancy including melanoma that expresses ESO as assessed by one of the following methods: reverse transcriptase-polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO Rapid influenza diagnostic test (RIDT), polymerase chain reaction (PCR), or viral culture positive for influenza Evaluable disease as demonstrated by clinical and/or radiologic studies with current or prior elevated blood levels of EBV-deoxyribonucleic acid (DNA) exceeding copies/ml by quantitative real time polymerase chain reaction (PCR), OR Patients with histologically proven glioblastomas or gliosarcomas that express EGFRvIII as assessed by immunohistochemistry (IHC) or polymerase chain reaction (PCR) confirmed by the National Cancer Institute (NCI) Laboratory of Pathology A diagnosis of APL based on the presence of the PML-RAR-alpha fusion gene by cytogenetics, polymerase chain reaction (PCR), or POD test The tumor must have been determined to be mismatch repair proficient or microsatellite stable through CLIA approved testing (Immunohistochemistry [IHC], polymerase chain reaction [PCR], or Next-Generation Sequencing [NGS] assays). Part patients must have tumor MGMT methylation status of unmethylated; results of routinely used methods for MGMT methylation testing (e.g. mutagenically separated polymerase chain reaction [MSPCR] or quantitative polymerase chain reaction [PCR]) are acceptable Participants with mature B-cell (Burkitts) ALL are excluded from study; mature B-cell is defined by the presence of surface immunoglobulin and/or the t(;)(q;q), t(;), or t(;) translocation and/or c-myc-gene rearrangement by fluorescence in situ hybridization (FISH) or polymerase chain reaction (PCR); (FISH/PCR testing for c-myc rearrangements is not required prior to study entry, but it is suggested for patients with surface immunoglobulin expression or L morphology) Patients must have tumor MGMT methylation status of unmethylated as determined by local pathologist using a Clinical Laboratory Improvement Act (CLIA)-approved diagnostic test; results of routinely-used methods for MGMT methylation testing (e.g. methylation-specific [MS]- polymerase chain reaction [PCR] or quantitative PCR) are acceptable Tissue available from primary and/or recurrent disease to evaluate tumor expression of NY-ESO- or PDL by immunohistochemistry (IHC) and/or reverse transcriptase-polymerase chain reaction (RT-PCR), and for measurement of DNA methylation Note: patients with mature B-cell (Burkitt's) ALL are excluded from study; mature B-cell is defined by the presence of surface immunoglobulin and/or the t(;)(q;q), t(;), or t(;) translocation and/or c-myc-gene rearrangement by fluorescence in situ hybridization (FISH) or polymerase chain reaction (PCR) Tumor MGMT methylation status must be available; results of routinely used methods for MGMT methylation testing (e.g. methylation-specific polymerase chain reaction [MSPCR ] or quantitative polymerase chain reaction [PCR]) are acceptable Documentation of HTLV infection (enzyme linked immunosorbent assay [ELISA]) in individual with confirmation of HTLV- infection (by immunoblot or polymerase chain reaction [PCR]) or a consistent clinical picture (including two of the following: ) CD+ leukemia or lymphoma, ) hypercalcemia, and/or ) Japanese, Caribbean or South American birthplace) Patients must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis Diagnosis of advanced stage or metastatic HER-positive cancer (immunohistochemistry or reverse transcriptase-polymerase chain reaction [RT-PCR] is used to determine HER positivity) Active Hepatitis B (by surface antigen expression or polymerase chain reaction) or C (by polymerase chain reaction) infection or on hepatitis-related antiviral therapy within months of first dose of study drug. HPV- negative SCCHN tumor as determined per institutional standard (eg, p IHC; multiplex nucleic acid sequence based amplification [NASBA] or other polymerase chain reaction [PCR]-based assays). Must have evidence of MET expression by fluorescence in situ hybridization (FISH), MET immunohistochemistry (IHC) score of -+, reverse-transcriptase polymerase chain reaction (RT-PCR) or a mutation MART- positive melanoma by reverse transcription (RT)-polymerase chain reaction (PCR) or immunohistochemistry (IHC) Detection of one of the following must be present:\r\n* t(;)(q;q) or -way variant by metaphase cytogenetics\r\n* Breakpoint cluster region (BCR)-Abelson (ABL) positive status by molecular analysis with qualitative polymerase chain reaction (PCR) or fluorescence in situ hybridization (FISH) Expanded cohort only: Cohort : patients with predominant metastatic bone disease; Cohort : patients with primary squamous head and neck cancers; Cohort : patients presenting any molecular abnormality of interest, which can include an ALK translocation, ALK amplification, ALK mutation and overexpression as determined by fluorescent in situ hybridization (FISH), immunohistochemistry (IHC), quantitative polymerase chain reaction (qPCR), quantitative reverse transcriptase-polymerase chain reaction (qRT-PCR), array Comparative Genomic Hybridization or direct sequencing (aCGH); a c-MET abnormality, either c-MET amplification by FISH, overexpression by IHC or c-MET mutation; BRAF, DDR and CDKNA mutations; and, finally, TRIM expression and CCN expression Participants must have histologically or cytologically confirmed squamous cell carcinoma of the oropharynx, unknown primary, or nasopharynx that is p positive as determined by immunohistochemistry (IHC); tissue from the primary site must be available for biomarker studies and for polymerase chain reaction (PCR) testing; IHC must be performed in a lab verified by the central laboratory or the slides must be available for review by the central laboratory (Zhang, Mount Sinai School of Medicine [MSSM]) and PCR must be done in the central laboratory prior to randomization; HPV PCR must be performed and results available for randomization after induction Participants must demonstrate evidence of persistent disease either by cytogenetics/FISH or by polymerase chain reaction (PCR) for BCR/ABL in the peripheral blood or bone marrow Measurable metastatic melanoma that expresses ESO as assessed by one of the following methods: reverse transcription polymerase chain reaction (RT-PCR) on tumor tissue, or by immunohistochemistry of resected tissue, or serum antibody reactive with ESO Positive NY-ESO- expression by reverse transcription (RT)-polymerase chain reaction (PCR) and/or immunohistochemistry (IHC) will be required for entry, as determined by analysis at the trial central laboratory Measurable metastatic or refractory/recurrent HPV-+ cancer (determined by in situ hybridization [ISH] or a polymerase chain reaction [PCR]-based test) Patients must have rearrangement involving the MLL gene, including reciprocal chromosomal translocations involving q by FISH, cytogenetic analysis, polymerase chain reaction (PCR) or next-generation sequencing (NGS) OR partial tandem duplication (PTD) of MLL by PCR or NGS. The patient's blasts cells show expression of WT transcript, detected by quantitative Reverse Transcription-Polymerase Chain Reaction (qRT-PCR).The patient received the following therapy according to the Institution's standard of care. Tumor expression of NY-ESO- by immunohistochemistry (IHC) and/or real-time polymerase chain reaction (RTPCR) Unresectable thyroid cancer expressing TG as assessed by one of the following methods: real-time (RT)-polymerase chain reaction (PCR) on tumor tissue, or by immunohistochemistry of resected tissue Must have HERV-K(HML) viral load of >= x ^ using a gag primer reverse transcriptase (RT)-polymerase chain reaction (PCR) assay Patients with active Hepatitis B or C viral replication by polymerase chain reaction (PCR) Patients with active hepatitis B or C determined by polymerase chain reaction (PCR) Determination of MGMT promoter status by methylation-specific polymerase chain reaction (PCR) per local institutional guidelines is required to assess eligibility for this Arm. Archived tissue from the CRC primary tumor in sufficient amounts to allow advanced quantitative real time-polymerase chain reaction (qRT-PCR) analysis; specimen from metastatic sites are not required but highly preferred Histologically or cytologically confirmed diagnosis of metastatic non-small cell lung cancer (NSCLC) (stage IV, American Joint Committee on Cancer [AJCC] v.) that carries (a) an ALK rearrangement, as determined by fluorescence in-situ hybridization (FISH), immunohistochemistry, or next generation sequencing (NGS) of either tissue or plasma, or (b) a ROS rearrangement as determined by FISH or reverse transcriptase-polymerase chain reaction (RT-PCR) or NGS via a local diagnostic test (LDT) or plasma analysis Serologic status and/or polymerase chain reaction (PCR) testing reflecting active hepatitis B or C infection