Inclusion Criteria- Core  Patient must be aged ≥ 18 years at the time of signing the informed consent. In some countries, parental consent may be required in addition to an assent form for patients who are 18 years of age. Patient must have histologically documented diagnosis of B-cell non-Hodgkin lymphoma (B-NHL) as defined by a B-cell neoplasm in the World Health Organisation classification scheme except as noted in the exclusion criteria. Patient has relapsed after or failed to respond to at least 2 but no more than 5 prior systemic treatment regimens (including investigational therapy) and for whom there is no available therapy expected to improve survival (eg, standard chemotherapy, autologous stem cell transplantation (SCT), chimeric antigen receptor T cell (CAR-T) cell therapy).  Documented active disease requiring treatment that is relapsed or refractory defined as:  Recurrence/relapse of disease after response to prior line(s) of therapy. Progressive disease (refractory) on/after completion of the treatment regimen preceding entry into the study.  Must have at least one measurable, fluorodeoxyglucose positron emission tomography (FDG-PET) avid lesion (except for MZL), based on bi-dimensional assessment on PET and computed tomography (CT)/magnetic resonance imaging (MRI) scan. A measurable lesion is defined as:  For nodal lesions: longest diameter > 1.5 cm For extranodal lesions: longest diameter > 1 cm Eastern Cooperative Oncology Group (ECOG) performance status score ≤ 2. Performance status must not have deteriorated by ≥ 2 levels within 2 weeks after providing informed consent. Adequate haematologic, hepatic, and renal function Adequate cardiac function as demonstrated by left ventricular ejection fraction > 50% on screening cardiac multigated acquisition, magnetic resonance imaging, or echocardiogram. Women of childbearing potential and men should use protocol defined contraceptive measures. Willing and able to participate in all required study evaluations and procedures including receiving IV administration of study intervention and admission to the hospital, when required, for administration of study treatment and monitoring. All patients must be willing to undergo an incisional or excisional lymph node or tissue biopsy or to provide a lymph node or tissue biopsy from the most recent available archival tissue. For inclusion in the genetic component of the study, patients must fulfil protocol defined criteria.  Inclusion Criteria- Module 1  Additional Inclusion Criteria for Cohort B1 (R/R mantle cell lymphoma [MCL]):  Histologically confirmed MCL, with documentation of monoclonal B cells that have a chromosome translocation t(11;14)(q13;q32) and/or overexpress cyclin D1, as assessed by Investigator or local pathologist. Must have relapsed after or failed to respond to at least 2 prior lines of treatment, including one anti-CD20 monoclonal antibody (mAb) and a Bruton's tyrosine kinase inhibitor.  Additional Inclusion Criteria for Cohort B2 (R/R FL or MZL):  Histologically confirmed diagnosis of FL Grade 1, 2, or 3a OR histologically confirmed MZL including splenic, nodal, and extranodal subtypes, as assessed by Investigator or local pathologist. For FL patients: Previously received at least 2 prior systemic treatment regimens (including anti-CD20 mAb and an alkylating agent). For MZL patients: Previously received at least 2 prior lines of systemic therapy including at least one anti-CD20 mAb-directed regimen either as monotherapy or as chemoimmunotherapy (Helicobacter pylori eradication and radiation therapy alone will not be considered a systemic treatment regimen).  Additional Inclusion Criteria for Cohort B3 (R/R DLBCL):  Histologically confirmed DLBCL (including transformed FL) OR FL Grade 3b. Must have received 2 lines of systemic therapy including at least one anti-CD20 mAb-directed regimen and must have failed or are ineligible for stem cell transplantation (if indicated per local institutional guidelines).  Exclusion Criteria- Core  Diagnosis of post-transplant lymphoproliferative disease, Richter's transformation, Burkitt's lymphoma, Burkitt-like lymphoma, lymphoblastic lymphoma/leukaemia, chronic lymphocytic leukaemia, small lymphocytic lymphoma. High risk of TLS according to Howard modification of Cairo-Bishop criteria and/or the presence of bulky disease. Unresolved toxicity from prior anticancer therapy. Patients with Grade 2 neuropathy or Grade 2 alopecia are eligible. Active idiopathic thrombocytopenic purpura. Active central nervous system (CNS) involvement by lymphoma, leptomeningeal disease or spinal cord compression. Known history of infection with human immunodeficiency virus. Known serologic status reflecting active hepatitis B or C infection; concurrent infection with cytomegalovirus (CMV). Patients must be tested for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and those with active infection in accordance with local testing guidelines will be excluded. Any evidence of severe or uncontrolled systemic diseases; current unstable or uncompensated respiratory or cardiac conditions; uncontrolled hypertension; history of, or active, bleeding diatheses; uncontrolled active systemic fungal, bacterial, or other infection. Any of the following cardiac criteria at screening: patients with a history of myocarditis within one year of study entry, or heart failure; mean resting corrected QT interval (QTcF) ≥ 470 msec obtained from 3 electrocardiograms (ECGs), in the absence of a cardiac pacemaker; any factors that increase the risk of QTc prolongation or risk of arrhythmic events; any clinically important abnormalities in rhythm, conduction or morphology of resting ECG. History of another life-threatening malignancy ≤ 2 years prior to first dose of study intervention. Any of the following currently or in the 6 months prior to the first dose of study intervention: coronary artery bypass graft; angioplasty; vascular stent; myocardial infarction; angina pectoris; haemorrhagic or thrombotic stroke, including transient ischaemic attacks or any other CNS bleeding. Treatment with any of the following: radiotherapy less than 2 weeks prior to the first dose of study intervention; any investigational agents or study drugs from a previous clinical study within ≤ 14 days or 5 half-lives prior to the first dose of study intervention; any other chemotherapy, immunotherapy, immunosuppressant medication or anticancer agents within 21 days of the first dose of study intervention; Prior allogenic haematopoietic stem cell transplantation (HSCT) within 6 months from the first dose of study intervention (patients > 6 months after allogenic HSCT are eligible in the absence of active graft-versus host disease and concomitant immune-suppressive therapy). Eligible patients must have stopped immunosuppression at least 2 months prior to study entry; prior cellular therapies such as CAR-T and/or autologous HSCT within 3 months prior to the first dose of study intervention; major surgery ≤ 21 days, or minor surgical procedures ≤ 7 days, prior to the first dose of study intervention; prescription or non-prescription drugs or other products known to be sensitive substrates of BCRP, OCT2, OAT3, OATP1B1, OATP1B3, CYP2B6, CYP2C8, CYP2C9 or CYP2D6, or reversible moderate or strong cytochrome 3A (CYP3A) inhibitors, which cannot be discontinued within 5 half-lives of the first dose of study intervention and withheld throughout the study until 14 days after the last dose of AZD0466; moderate or strong mechanism-based inhibitors or inducers of CYP3A4 which cannot be discontinued within 5 half-lives plus 12 days of the drug prior to the first dose of study intervention and withheld until 14 days after the last dose of AZD0466; concurrent anticoagulation therapy, including aspirin, which cannot be stopped; medications with known risk of Torsades de Pointes within 5 half-lives of the first dose of study intervention and continuing until 5 half-lives after the last dose of AZD0466. Administration of a live, attenuated vaccine within 4 weeks before first dose of study intervention. Administration of inactivated vaccines or protein/RNA immunogen vaccines. Patients with a known hypersensitivity to polyethylene glycol, pegylated products, or drugs with a similar chemical structure or class to AZD0466 or other BH3 mimetic.  Exclusion Criteria- Module 1  Additional Exclusion Criteria for Cohort B1:  - Patients with known blastoid or pleiomorphic variant at study entry/most recent relapse.  Additional Exclusion Criteria for Cohort B2:  Histologically confirmed diagnosis of FL grade 3B. Known transformation to aggressive lymphoma, eg, large cell lymphoma.