Inclusion Criteria:  All Cohorts:  Must have an Eastern Cooperative Oncology Group (ECOG) performance status score 0 - 2.  Has one or more measurable disease sites:  Fluorodeoxyglucose-positron emission tomography (FDGPET) scan demonstrating positive lesion compatible with computed tomography (CT) or magnetic resonance image (MRI)-defined anatomical tumor sites. >= 1 measurable nodal lesion (long axis >= 1.5 cm and short axis > 1.0 cm) or >= 1 measurable extra-nodal lesion (long axis >= 1 cm) on CT scan or MRI.  Cohort 1 Part 1 (Monotherapy Safety Run-in) Specific Criteria:  Must have histologically confirmed CD20+ Diffuse large B-cell lymphoma (DLBCL), or High-grade B-cell lymphoma (HGBL) with MYC and BCL2 and/or BCL6 rearrangements and DLBCL feature, and follicular Lymphoma (FL) at most recent (previous or current) representative tumor biopsy based on the pathology report, according to the World Health Organization (WHO) 2016 classification. Must have at least one prior treatment with an anti-CD20 monoclonal antibody (e.g., rituximab) potentially in combination with chemotherapy and/or relapsed after autologous stem cell rescue. Must have relapsed or refractory disease.  Cohort 1 Part 2 (Monotherapy Expansion) Specific Criteria:  Must have histologically confirmed CD20+ DLBCL (de novo or transformed) at most recent (previous or current) representative tumor biopsy based on the pathology report, according to the World Health Organization (WHO) 2016 classification Following safety run-in and up to the 12th participant (including the number of safety run-in participants) in Cohort 1, participant must have received at least 1 prior line of systemic therapies which must include an anti-CD20 monoclonal antibody containing combination therapy (e.g., rituximab). After 12 participants have been enrolled at dose A of epcoritamab, participants must have received at least 2 prior lines of systemic therapies. Either failed prior autologous HSCT, or ineligible for autologous HSCT due to age or comorbidities.  Cohort 2 Specific Criteria:  Must have newly diagnosed DLBCL.  Double-/triple-hit DLBCL (categorized in WHO 2016 as high-grade B-cell lymphoma [HGBCL] with MYC and BCL2 and/or BCL6 rearrangements)*  * Note: other histologies with MYC and BCL2 and/or BCL6 translocations are excluded.  Eligible for standard R-CHOP for 6 cycles.  Cohort 3 Specific Criteria:  Must have histologically confirmed Grade 1 - 3a Follicular Lymphoma with no evidence of histologic transformation to an aggressive lymphoma at most recent (previous or current) representative tumor biopsy and CD20+ on a representative tumor biopsy based on the pathology report, according to the WHO 2016 classification. Must have R/R disease to at least one prior treatment which must include an anti-CD20 monoclonal antibody (e.g., rituximab). Must be eligible for R2 per investigator determination.  Exclusion Criteria:  All Cohorts:  - History of primary mediastinal lymphoma.  Cohort 2 Specific Criteria:  - History of prior systemic anti-lymphoma therapy (including definitive radiotherapy) for Diffuse large B-cell lymphoma (DLBCL) other than corticosteroids.