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):

          1. Histologically confirmed diagnosis of FL Grade 1, 2, or 3a as assessed by investigator
             or local pathologist

          2. Current need for systemic treatment based on the Investigator's opinion

          3. 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)

          4. Participants should have received up to a maximum of 5 lines of prior therapies.

          5. 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):

          1. Histologically confirmed MZL including splenic, nodal, and extranodal subtypes as
             assessed by investigator or local pathologist

          2. Current need for systemic treatment based on the Investigator's opinion

          3. 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)

          4. Participants should have received up to a maximum of 5 lines of prior therapies

          5. 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):

          1. 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

          2. Relapsed, progressed or refractory (defined as failure to achieve at least a PR) after
             at least 2 prior systemic lines of therapy

          3. 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

          4. 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:

               1. Absolute neutrophil count < 1.0 × 10^9/L; < 0.75 × 10^9/L in participants with
                  known bone marrow involvement of malignant disease

               2. Platelets < 75 × 10^9/L; < 50 × 10^9/L in participants with known bone marrow
                  involvement of malignant disease

               3. 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:

               1. 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

               2. 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

               3. 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

               4. Receipt of live, attenuated vaccine within 28 days before the first dose of study
                  treatment(s)

               5. 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:

          1. Follicular lymphoma grade 3B

          2. Known transformation to aggressive lymphoma, eg, large cell lymphoma

          3. MCL participants with blastoid or pleiomorphic variant or documented tumour protein 53
             mutation at study entry/most recent relapse

          4. Participants who, in the Investigator's opinion, require immediate cytoreductive
             therapy for disease control