Inclusion Criteria:  Participants must be ≥ 18 years of age, at the time of signing the informed consent Eastern Cooperative Oncology Group performance status ≤ 2 Life expectancy > 6 months Female participants must not be breast-feeding and must have a negative pregnancy test prior to start of dosing  Module 1 specific inclusion criteria:  Additional Inclusion Criteria for Cohort 1A (R/R FL):  Histologically confirmed diagnosis of FL Grade 1, 2, or 3a as assessed by investigator or local pathologist Current need for systemic treatment based on the Investigator's opinion Relapsed, progressed or refractory (defined as failure to achieve at least a partial response [PR]) after at least 2 prior systemic lines of therapy (including anti-CD20mAb and an alkylating agent) Participants should have received up to a maximum of 5 lines of prior therapies. Bi-dimensionally measurable disease on cross sectional imaging by computed tomography (CT) or magnetic resonance imaging (MRI) with at least one nodal lesion > 1.5 cm in the long axis or at least one extranodal lesion > 1 cm in long axis.  Additional Inclusion Criteria for Cohort 1B (R/R MZL):  Histologically confirmed MZL including splenic, nodal, and extranodal subtypes as assessed by investigator or local pathologist Current need for systemic treatment based on the Investigator's opinion Relapsed, progressed or refractory (defined as failure to achieve at least a PR) after at least 2 prior systemic lines of therapy (including at least one anti-CD20mAb directed regimen either as monotherapy or as chemoimmunotherapy; Helicobacter pylori eradication and radiation therapy alone will not be considered a systemic treatment regimen) Participants should have received up to a maximum of 5 lines of prior therapies Bi-dimensionally measurable disease on cross sectional imaging by CT or MRI with at least one nodal lesion > 1.5 cm in the long axis or at least one extranodal lesion > 1 cm in long axis  Additional Inclusion Criteria for Cohort 1C (R/R 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 Relapsed, progressed or refractory (defined as failure to achieve at least a PR) after at least 2 prior systemic lines of therapy  Participants should have received up to a maximum of 4 lines of prior therapies  Prior regimens must have included:  BTK inhibitor and Anti-CD20 monoclonal antibody therapy Bi-dimensionally measurable disease on cross sectional imaging by CT or MRI with at least one nodal lesion > 1.5 cm in the long axis or at least one extranodal lesion > 1 cm in long axis  Exclusion Criteria:  Prior malignancy (other than the disease under study), except for adequately treated basal cell or squamous cell skin cancer, in situ cancer, or other cancer from which the participant has been disease free for ≥ 2 years With the exception of alopecia, any unresolved non-haematological toxicities from prior therapy ≥ Common Terminology Criteria for Adverse Events Grade 2 at the time of starting study treatment Known medically apparent central nervous system lymphoma or leptomeningeal disease  Inadequate bone marrow reserve or organ function as demonstrated by any of the following laboratory values at screening:  Absolute neutrophil count < 1.0 × 10^9/L; < 0.75 × 10^9/L in participants with known bone marrow involvement of malignant disease Platelets < 75 × 10^9/L; < 50 × 10^9/L in participants with known bone marrow involvement of malignant disease Creatinine clearance < 50 mL/min per the Cockcroft and Gault formula Clinically significant abnormalities of glucose metabolism as participants with diabetes mellitus type I or diabetes mellitus type II requiring insulin treatment and Glycosylated haemoglobin ≥ 8.0% (63.9 mmol/mol)  Prior treatment with any of the following:  Any investigational agents or study drugs from a previous clinical study within 5 half lives or 2 weeks from the first dose of capivasertib in this study Potent inhibitors or inducers of CYP3A4 and/or UGT2B7 within 2 weeks prior to the first dose of study treatment (3 weeks for St John's wort), or sensitive substrates of CYP3A4, CYP2C9 and/or CYP2D6 with a narrow therapeutic window within 1 week prior to the first dose of study treatment Prior allogenic Haematopoietic stem cell transplant (HSCT) within 6 months from the first dose of capivasertib (patients > 6 months after allogenic HSCT are eligible in the absence of active graft-versus-host disease and concomitant immune suppressive therapy). Prior cellular therapies (eg, Chimeric antigen receptor T therapy) and/or autologous HSCT within 3 months from the first dose of capivasertib Receipt of live, attenuated vaccine within 28 days before the first dose of study treatment(s) Participants who, due to other medical conditions /prior history /concomitant medications are, in the investigator's opinion, at a risk of a venous thromboembolism (VTE) and are not willing to accept the VTE prophylaxis, will be excluded. The initiation of an adequate VTE prophylaxis will be based on treating physician risk/benefit assessment and in agreement with the local management guidelines  Additional exclusion core criteria may apply, please refer to the protocol  Module 1 specific exclusion criteria:  Follicular lymphoma grade 3B Known transformation to aggressive lymphoma, eg, large cell lymphoma MCL participants with blastoid or pleiomorphic variant or documented tumour protein 53 mutation at study entry/most recent relapse Participants who, in the Investigator's opinion, require immediate cytoreductive therapy for disease control