Study to Evaluate the Safety, Tolerability, Pharmacokinetics, and Efficacy of AMG 562 in Subjects With r/r Diffuse Large B-cell Lymphoma, Mantle Cell Lymphoma, or Follicular Lymphoma

Inclusion Criteria: * Subject has provided informed consent prior to initiation of any study-specific activities/procedures * Age ≥ 18 at the time of informed consent. * Biopsy proven B-NHL including: * DLBCL, which also includes DLBCL that represents transformation of indolent NHL (including follicular, marginal zone, and lymphoplasmacytic lymphoma excluding chronic lymphocytic leukemia or Hodgkin Lymphoma) and DLBCL with alterations of MYC and BCL2 and/or BCL6 also described as double-hit and triple-hit lymphomas. * FL * MCL Presentations of these histologies with substantial occurrence of malignant cells into the bloodstream (lymphocyte count ≥ 7 x 10\^9/L) including all leukemic presentations are excluded. The following histologies are not eligible: Lymphoblastic lymphoma Burkitt lymphoma Any histologies not specifically mentioned must be discussed with the Medical Monitor * Subjects with transformation of indolent lymphoma must have received therapy after a diagnosis of transformation that is appropriate for aggressive histology. \- Subjects who received prior CD19-targeting treatment are allowed (CAR-T cell therapy is excluded). A biopsy following CD19-targeting treatment is required unless no lesions are accessible or the risk of the biopsy is deemed too high by the investigator For Part 2 (Expansion in patients with DLBCL): only biopsy proven DLBCL (biopsy proven at least at primary diagnosis), including DLBCL that represents transformation of indolent NHL (including follicular, marginal zone, and lymphoplasmacytic lymphoma excluding chronic lymphocytic leukemia or Hodgkin Lymphoma) are eligible. Other histologies are not eligible. * Presentations of these histologies with substantial occurrence of malignant cells into the bloodstream (lymphocyte count ≥ 7 x 10\^9/L) including all leukemic presentations are excluded. * Subjects with transformation of indolent lymphoma must have received therapy after a diagnosis of transformation that is appropriate for aggressive histology as described in inclusion criterial. * Subjects who received prior CD19-targeting treatment are allowed (CAR-T cell therapy is excluded) A biopsy following CD19-targeting treatment is required unless no lesions are accessible or the risk of the biopsy is deemed too high by the investigator - For DLBCL: Refractory (no prior CR/CMR) to first or later line of treatment or relapsed (prior CR/CMR) after two or more prior treatments, with at least one treatment consisting of standard multiagent chemotherapy containing an anthracycline AND an approved anti-CD20 agent. Examples of appropriate therapy include but are not limited to R-CHOP (14 or 21), R-CHOEP, and DA-R-EOCH. For FL: Refractory (no prior CR/CMR) to first or later line of treatment or relapsed (prior CR/CMR) after three or more prior treatments, with at least one treatment consisting of a standard chemotherapy containing an approved anti-CD20 agent. Examples of appropriate therapy include but are not limited to R-CHOP, R-CVP, and BR. For MCL: Refractory (no prior CR/CMR) to first or later line of treatment or relapsed (prior CR/CMR) after three or more prior treatments, with at least one treatment consisting of a standard chemotherapy containing an approved anti-CD20 agent. Examples of appropriate therapy include but are not limited to R-CHOP, BR and hyper-CVAD alternating with R-MTX/Ara-C. For subjects with refractory B-NHL and who have received radiotherapy, PET positivity should be demonstrated no less than 6 weeks after the last dose of radiotherapy * Minimum life expectancy of 12 weeks * Radiographically measurable disease with a clearly demarcated nodal lesion at least 1.5 cm in its largest dimension or a target extranodal lesion at least 1.0 cm in its largest dimension. In the dose exploration phase in case disease is not radiographically measurable PET positivity (ie, Deauville ≥4) instead is acceptable. * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Laboratory parameters (completed within 14 days prior to enrollment): Hematology: * Absolute neutrophil count (ANC) ≥ 1.0 x 10\^9/L * Platelets ≥ 75 x 10\^9/L Chemistry: * Creatinine clearance ≥ 60 mL/min (calculated using Cockcroft Gault equation) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \< 3X upper limit of normal (ULN) * Total bilirubin (TBL) \< 2x ULN (unless Gilbert's disease or if liver involvementwith lymphoma) Exclusion Criteria: * Treatment within 30 days prior to enrollment with another investigational device or drug (interventional clinical study / studies). Other investigational procedures while participating in this study are excluded (observational studies are permitted). * Prior anti-cancer therapy as specified below: * At least 6 weeks must have elapsed since any prior systemic inhibitory/stimulatory immune checkpoint molecule therapy (eg, ipilimumab, nivolumab, pembrolizumab, atezolizumab, OX40 agonists, 4-1BB agonists, etc) before the first dose of AMG 562. * Other targeted anti-cancer therapy (chemotherapy, molecular targeted therapy, steroids) within 14 days or 5 half lives (which ever is longer) prior to first dose of AMG 562. Patients requiring continued treatment due to aggressive disease may only be included if there is agreement by both the investigator and the Amgen Medical Monitor. * Radiation therapy completed within 28 days prior to first dose of AMG 562. * Autologous HSCT within six weeks prior to start of AMG 562 treatment. * At least 4 weeks must have elapsed since any prior treatment with antibody therapy (exception immune checkpoint inhibitors) before the first dose of AMG 562. * Prior CD19-directed CAR-T cell therapies * Prior allogeneic HSCT. * For Part 2 (Expansion in patients with DLBCL): fluorodeoxyglucose non-avid patients. * Baseline electrocardiogram (ECG) QTc \> 470 msec. * Autoimmune disorders requiring chronic systemic steroid therapy or any other form of immunosuppressive therapy. Patient may be included if the treatment is discontinued more than 3 months prior to the first dose of AMG 562 at a low likelihood of relapse AND if there is agreement by both the investigator and the Amgen Medical Monitor. * Unresolved toxicity from prior anti-tumor therapy, defined as not having resolved to CTCAE version 4.0 grade 1, or to levels dictated in the eligibility criteria with the exception of alopecia or toxicities from prior anti-tumor therapy that are considered irreversible (defined as having been present and stable for \> 28 days) which may be allowed if they are not otherwise described in the exclusion criteria AND there is agreement to allow by both the investigator and the Amgen Medical Monitor. * Presence of clinically relevant central nervous system (CNS) pathology such as epilepsy, paresis, aphasia, stroke, severe brain injury, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis. * Evidence of CNS involvement by NHL. * Known infection with human immunodeficiency virus (HIV). * Exclusion of hepatitis infection based on the following results and/or criteria: * Positive for hepatitis B surface antigen (HBsAg) (indicative of chronic hepatitis B or recent acute hepatitis B). * Negative HBsAg and positive for hepatitis B core antibody: Assay for hepatitis B virus DNA by polymerase chain reaction (PCR) is necessary. Detectable hepatitis B virus DNA suggests occult hepatitis B. * Positive Hepatitis C virus antibody (HCVAb): Assay for hepatitis C virus RNA by PCR is necessary. Detectable hepatitis C virus RNA suggests chronic hepatitis C. * History of malignancy other than B-NHL within the past 3 years with the exception of: * Malignancy treated with curative intent and with no known active disease present for ≥ 3 years before enrollment and felt to be at low risk for recurrence by the treating physician. * Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease. * Adequately treated cervical carcinoma in situ without evidence of disease. * Adequately treated breast ductal carcinoma in situ without evidence of disease. * Prostatic intraepithelial neoplasia without evidence of prostate cancer. * Adequately treated urothelial papillary noninvasive carcinoma or carcinoma in situ. * Major surgery within 28 days of first dose AMG 562. * History of arterial thrombosis (eg, stroke or transient ischemic attack) within 12 months of first dose of AMG 562. * Presence of fungal, bacterial, viral, or other infection requiring IV antimicrobials for management within 7 days of first dose AMG 562. NOTE: Simple UTI and uncomplicated bacterial pharyngitis are permitted after consultation with sponsor and if responding to active treatment. * Subject has known sensitivity to immunoglobulins or any of the products or components to be administered during dosing. * Males and females of reproductive potential who are unwilling to practice highly effective method(s) of birth control while on study through 110 days (females) and 170 days (males) after receiving the last dose of study drug. Highly effective methods of birth control include sexual abstinence (males, females); vasectomy; bilateral tubal ligation/occlusion; or a condom with spermicide (males) in combination with hormonal birth control or intrauterine device (IUD) (females). * Females who are lactating/breastfeeding or who plan to breastfeed while on study through 110 days after receiving the last dose of study drug. * Females with a positive pregnancy test. * Females planning to become pregnant while on study through 110 days after receiving the last dose of study drug. * Males who are unwilling to abstain from sperm donation while on study through 170 days after receiving the last dose of study drug. * Subjects likely to not be available to complete all protocol- required study visits or procedures including BM aspirates/biopsies, and/or to comply with all required study procedures to the best of the subject and investigator's knowledge. * History or evidence of any other clinically-relevant concurrent disorder, condition or disease (eg, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia requiring therapy at time of screening) with the exception of those outlined above that, in the opinion of the investigator or Amgen medical monitor, if consulted, would not pose a risk to subject safety or interfere with the study evaluation, procedures or completion.