Study of Pembrolizumab With Chemotherapy in Patients With Advanced Lymphoma (PembroHeme)

Inclusion Criteria: 1. Patient at least 18 years old and has definitive histologically or cytologically confirmed: classical Hodgkin lymphoma (HL) or diffuse large B-Cell lymphoma (DLBCL) non-Hodgkin lymphoma (NHL). 2. Patient has one or more metastatic lesions \>1.5 cm as defined by lymphoma response criteria by PET-CT scan. If CT scan is needed for clarification of findings on PET-CT it may be done additionally. Tumor sites that are considered measureable must not have received prior radiation therapy. 3. Patients can be enrolled only on one of the treatment arms on this trial. 4. The investigator will select the appropriate treatment arm for the patient with the following requirements: (a) Patients cannot have had prior progression or intolerance on the single agent chemotherapy and then enrolled on an arm with that same single agent chemotherapy plus pembro (b) The chemotherapy on the arm selected must be considered standard of care or listed in www.nccn.org for that cancer type. 5. Have recovered from acute toxicities of prior treatment: * \> 3 weeks must have elapsed since receiving any investigational agent. * \> 2 weeks must have elapsed since receiving any radiotherapy, or ≥ 3 weeks or 5 half-lives whichever is shorter for treatment with cytotoxic or biologic agents ( ≥ 6 weeks for mitomycin or nitrosoureas). Chronic treatment with non-investigational gonadotropin-releasing hormone analogs or other hormonal or supportive care is permitted. 6. Patient has adequate biological parameters as demonstrated by the following blood counts at time of screening: 7. Absolute neutrophil count (ANC) \> 1000/ microliter, platelet count ≥ 75,000/ microliter, hemoglobin ≥ 9 g/dL. Subject can be given packed red blood cell transfusion. 8. Calculated creatinine clearance \> 50 ml/min by Cockroft-Gault equation, total bilirubin 1.5 times the upper limit of normal (ULN) range, AST/ALT ≤ 3 times the upper limit of normal (ULN) range. 9. Thyroid stimulating hormone (TSH) within institutional normal limits. If TSH is above the upper limit of normal range, then a free T4 within institutional normal limits is acceptable. 10. At least 100 days must have elapsed in subjects that had a prior autologous transplant. 11. Resolution of prior systemic therapy non-hematologic AE to grade ≤ 2 (except alopecia or correctable electrolyte abnormality with supplementation) 12. Patient has a Karnofsky performance status (KPS) ≥ 60. 13. Women of child-bearing potential (i.e., women who are pre-menopausal or not surgically sterile) must be willing to use an acceptable contraceptive method (abstinence, oral contraceptive or double barrier method) for the duration of the study and for 4 months following the last dose of Pembrolizumab, brentuximab vedotin, or for 2 years following the last dose of chemotherapy on this trial, and must have a negative urine or serum pregnancy test within 2 weeks prior to beginning treatment on this trial. If a female subject or female partner of a male subject becomes pregnant during this period then patient will be recommended to seek appropriate obstetric care. The study will not be monitoring subjects or female partners of subjects for pregnancy after the last dose of study drug or chemotherapy. Exclusion Criteria: 1. Active clinically serious infection \> Grade 2. 2. Serious non-healing wound, ulcer, or bone fracture. 3. Patient has known brain metastases. 4. Patient has with myelodysplasia, polycythemia vera, idiopathic thrombocythemia, myelofibrosis, or acute leukemia. 5. Inability to complete informed consent process and adhere to the protocol treatment plan and follow-up requirements. 6. Patient has known active infection with HIV, hepatitis B, or hepatitis C (patients will be required to be tested for the presence of such viruses prior to therapy on this protocol). 7. Ejection fraction measuring less than 45% (measurement of ejection fraction is required only for subjects with prior anthracycline exposure). 8. Requiring daily corticosteroid dose ≥ 10 mg prednisone or equivalent per day. 9. Patient has undergone major surgery, other than diagnostic surgery (e.g., surgery done to obtain a biopsy for diagnosis without removal of an organ), within 4 weeks prior to Day 1 of treatment in this study. 10. Patient has a history of allergy or hypersensitivity to any of the study drugs or any of their excipients, or the patient exhibits any of the events outlined in the Contraindication or Special Warnings and Precautions sections of the product or comparator SmPC or Prescribing Information. 11. Patient has serious medical risk factors involving any of the major organ systems, or serious psychiatric disorders, which could compromise the patient's safety or the study data integrity. 12. Patient will be receiving any other anti-cancer therapy during participation in this trial. 13. Prior treatment with Pembrolizumab is not allowed. Prior receipt of other PD-1 inhibitors or PD-L1 inhibitors is allowed. 14. Active or prior documented autoimmune disease requiring systemic treatment within the past 2 years.