Inclusion Criteria:

        For Part 1 and Part 2 (osteosarcoma and neuroblastoma expansion cohorts):

          -  Participant is a child or an adolescent greater than or equal to (>=) 6 months to less
             than (<) 18 years old at the time of informed consent/assent.

          -  Participant will receive niraparib tablet or age-appropriate oral liquid formulation
             based on body weight and ability to swallow tablet.

          -  Performance status must be >=60 percent on the Karnofsky scale for participants >16
             years of age and >=60 percent on the Lansky scale for participants less than or equal
             to (<=) 16 years of age.

          -  Participant has adequate organ function.

          -  A female participant is eligible to participate if she is not pregnant or
             breastfeeding, and at least one of the following conditions applies: Is not a woman of
             childbearing potential (WOCBP) or Is a WOCBP and using a contraceptive method that is
             highly effective.

          -  A male participant of reproductive potential is eligible to participate if he agrees
             to refrain from donating sperm plus, either be abstinent from heterosexual intercourse
             or must agree to use a male condom.

        For Part 1 only:

          -  Participant has recurrent or refractory osteosarcoma, neuroblastoma, adrenocortical
             carcinoma, Ewing sarcoma, rhabdomyosarcoma, or any other solid tumor (excluding tumors
             of the central nervous system [CNS]) participants with non-CNS solid tumours other
             than osteosarcoma, neuroblastoma, adrenocortical carcinoma, Ewing sarcoma or
             rhabdomyosarcoma are required to have prior documented breast cancer susceptibility
             gene (BRCAness) mutational signature (mutational signature 3) on deoxyribonucleic acid
             (DNA) sequencing of tumor obtained in the relapsed/recurrent disease setting, within 6
             (preferably 3) months prior to enrolment and must not be eligible for local curative
             treatment. For participants with documented BRCAness mutational signature: Existing
             information on molecular profiling of the participant's tumor tissue must be through a
             molecular profiling platform such as Individualized Therapy for Relapsed Malignancies
             in Childhood (INFORM). Molecular profile information must contain information from
             whole exome sequencing or whole genome sequencing, including the mutation status of
             BRCA1/2 and other homologous recombination DNA repair (HRR) pathway genes, mutational
             signatures including signature 3, and tumor mutational burden (TMB).

        For Part 2 (osteosarcoma expansion cohort) only:

          -  Participant has recurrent or refractory osteosarcoma and must not be eligible for
             local curative treatment.

          -  Participant has radiographically measurable disease that can be tracked as RECIST v1.1
             target lesion(s).

          -  Participant must provide tumor tissue sample at screening for retrospective
             exploratory biomarker analysis.

        For Part 2 (neuroblastoma expansion cohort) only:

          -  Participant has recurrent or refractory neuroblastoma and must not be eligible for
             local curative treatment.

          -  Participant has radiographically measurable disease at the time of study enrolment;
             participants with neuroblastoma who do not have measurable disease but have
             metaiodobenzylguanidine-positive (+) evaluable disease are eligible. Measurable
             disease in participants with CNS involvement is defined as a tumor that is measurable
             in 2 perpendicular diameters on magnetic resonance imaging (MRI) and visible on more
             than 1 slice.

          -  Participant must provide tumor tissue sample at screening.

        Exclusion Criteria:

        For Part 1 and Part 2 (osteosarcoma and neuroblastoma expansion cohorts):

          -  Participant has known hypersensitivity to dostarlimab or niraparib, their components,
             or their excipients.

          -  Participant has a known history of myelodysplastic syndrome (MDS) or acute myeloid
             leukemia (AML).

          -  Participant has active autoimmune disease that has required systemic treatment in the
             past 2 years (that is [i.e.], with use of disease-modifying anti-rheumatic drugs,
             corticosteroids, or immunosuppressive drugs). Replacement therapy (for example [e.g.],
             thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or
             pituitary insufficiency) is not considered a form of systemic treatment.

          -  Participant has known active CNS metastases, carcinomatous meningitis, or both.
             Carcinomatous meningitis precludes a participant from study participation regardless
             of clinical stability.

          -  Participant had a known additional malignancy that progressed or required active
             treatment within the last 2 years.

          -  Participant is considered a poor medical risk due to a serious, uncontrolled medical
             disorder, non-malignant systemic disease, or active infection that requires systemic
             therapy.

          -  Participant has a condition (such as transfusion-dependent anemia or
             thrombocytopenia), therapy, or laboratory abnormality that might confound the study
             results or interfere with the participant's participation for the full duration of the
             study treatment including the following: Participants who received a transfusion
             (platelets or red blood cells) within 6 weeks of the first dose of study treatment are
             not eligible. Participants who received colony-stimulating factors (e.g.,
             granulocyte-colony stimulating factor [G-CSF], granulocyte-macrophage
             colony-stimulating factor, or recombinant erythropoietin) within 4 weeks prior to the
             first dose of study treatment are not eligible.

          -  Participant has a diagnosis of immunodeficiency or is receiving systemic steroid
             therapy or any other form of immunosuppressive therapy within 7 days prior to the
             first dose of study treatment.

          -  Participant has a known history of human immunodeficiency virus (HIV) (type 1 or 2
             antibodies).

          -  Participant has known active hepatitis B (e.g., hepatitis B surface antigen reactive)
             or hepatitis C (e.g., hepatitis C virus ribonucleic acid [qualitative] is detected).

          -  Participant has had any known Grade 3 or 4 anemia, neutropenia, or thrombocytopenia
             due to prior chemotherapy that persisted >4 weeks related to the most recent prior
             treatment.

          -  Participant had treatment with prior systemic anticancer therapy within the 3 weeks
             prior to the first dose of study treatment, radiation therapy encompassing >20 percent
             of the bone marrow within 2 weeks prior to the first dose of study treatment, or any
             radiation therapy within 1 week prior to the first dose of study treatment.

          -  Participant has received a live vaccine within 30 days of planned start of study
             treatment.

          -  Participant has clinically significant cardiovascular disease (e.g., significant
             cardiac conduction abnormalities, uncontrolled hypertension, cardiac arrhythmia or
             unstable angina, New York Heart Association Grade 2 or greater congestive heart
             failure, serious cardiac arrhythmia requiring medication, and history of
             cerebrovascular accident) within 6 months of enrolment.

          -  Participant has heart rate-corrected QT interval prolongation >480 milliseconds at
             screening.

          -  For Part 2 (osteosarcoma expansion cohort and neuroblastoma expansion cohort):

          -  Participant has received prior therapy with an anti-PD-1, anti-programmed cell death
             ligand 1, anti-programmed cell death-ligand 2, anti-cytotoxic T-lymphocyte-associated
             antigen-4 antibody (including ipilimumab), or any other antibody or drug specifically
             targeting T-cell co-stimulation or checkpoint pathways (with the exception of
             participants rolling over from Part 1 of the study: these participants are allowed to
             have received dostarlimab).

          -  Participant has had prior treatment with a known poly(adenosine diphosphate-ribose)
             polymerase (PARP) inhibitor (with the exception of participants rolling over from Part
             1 of the study: these participants are allowed to have received niraparib).