This research is being done to find out if an investigational drug, Nivolumab, can be safely administered after a "half-matched" (haplo) bone marrow transplant (BMT), and if the investigational drug will help to prevent or delay relapse or progression of sarcomas. In this study investigators will also be trying to learn more about how the investigational drug changes blood and/or tumors. Participants are eligible for this trial if they have recently undergone a "half-matched" (haplo) bone marrow transplant and have either relapsed or are at high risk to relapse.

High risk, recurrent, or refractory solid tumors in pediatric, adolescent and young adult (AYA) patients have an extremely poor prognosis despite current intensive treatment regimens. Johns Hopkins piloted an allogeneic bone marrow transplant (alloBMT) platform using a reduced intensity conditioning (RIC) and partially HLA-mismatched (haploidentical) related donors for this population of pediatric and AYA solid tumor patients.With this strategy, investigators demonstrated that RIC haploBMT with post-transplant cyclophosphamide (PTCy) is feasible and has acceptable toxicities in patients with incurable pediatric and AYA solid tumors; thus, this approach serves as a platform for post-transplant strategies to prevent relapse and optimize progression free survival. In this trial, the central hypothesis is that the efficacy of alloBMT for high risk solid tumors can be improved by developing methods to augment donor T cell responses against antigens selectively or uniquely expressed by tumor tissue.

Investigators aim to demonstrated that Programmed death-ligand 1 (PD-1) blockade with nivolumab will be safe and well tolerated after RIC haplo BMT, initially in a relapsed population (Part A) and ultimately when given pre-emptively (Part B).

Inclusion Criteria:

1. Patients must be ≥ 12 months and ≤ 50 years of age at the time of study enrollment.

2. Patients with histologically confirmed solid tumors with an estimated poor long term survival.

3. Performance Level: Karnofsky ≥ 50% for patients > 16 years of age and Lansky ≥ 60 for patients ≤16 years of age.

4. Patients must be post RIC haploidentical BMT.

5. Patients must have fully recovered from the acute toxic effects of prior BMT.

6. Concomitant radiation therapy can be administered in the setting of this trial.

7. Subjects must consent to allow for a baseline tumor biopsy. If a biopsy is not feasible, then archival tumor material must be made available. Tumor biopsies to be taken (if a subject's tumor is thought to be reasonably safe and easy to biopsy) at baseline (any time prior to the first dose after eligibility is met) and at Cycle 2 (4-6 cores per time point) or when lesions are visualized on physical examination or imaging studies in the case of no identifiable masses at cycle 2. Additional optional biopsies may be obtained later in the course of study treatment. The proposed investigation is considered a non-significant risk (NSR). A significant risk procedure is generally considered to be one for which the procedure-associated absolute risk of mortality or major morbidity, in the patient's clinical setting and at the institution completing the procedure, is 2% or higher. Diagnostic Tissue Samples Tissue, fluid, or blood may be collected from standard of care procedures used to treat or diagnose immune related toxicities/GVHD.

8. Organ Function Requirements:

   I. Adequate Hematologic Parameters:

   1. For patients with solid tumors without known bone marrow involvement:

      • Peripheral absolute neutrophil count (ANC) ≥ 500/mm3
      • Platelet count ≥ 50,000/mm3

   2. Patients with known bone marrow metastatic disease will be eligible for study without the above criteria. They may receive transfusions provided they are not known to be refractory to red cell or platelet transfusions. These patients will not be evaluable for hematologic toxicity.

   II. Adequate Renal Function Defined as:

   1. Creatinine clearance or radioisotope Glomerular filtration rate (GFR) ≥ 70ml/min/1.73 m2 or

   2. A serum creatinine based on age/gender as follows:

      • Age 1 to <2 years, Male: 0.6 and Female: 0.6
      • Age 2 to <6 years, Male: 0.8 and Female: 0.8
      • Age 6 to <10 years, Male: 1 and Female:1
      • Age 10 to <13 years, Male: 1.2 and Female 1.2
      • Age 13 to <16 years, Male: 1.5 and Female 1.4
      • Age ≥ 16 years, Male: 1.7 and Female 1.4

   III. Adequate Liver Function Defined as:

   1. Bilirubin (sum of conjugated + unconjugated) ≤1.5 x upper limit of normal (ULN) for age
   2. Serum glutamic pyruvic transaminase (SGPT) (ALT) ≤110 U/L. For the purpose of this study, the ULN for SGPT is 45 U/L.

9. Patients must have been registered on protocol J12106 "A Phase II Trial of Reduced Intensity Conditioning and HLA-matched or Partially HLA-mismatched (HLA-haploidentical) Related Donor Bone Marrow Transplant for High-risk Solid Tumors" before enrolling on this study.Patient may be screened prior to Day +120 but first dose of study drug must be given on or after Day +120.

Exclusion Criteria:

1. GVHD: any history of Stage 4 skin GVHD or Stage 3 gut/liver GVHD (a.k.a. overall Grade III/IV GVHD) or any severe chronic GVHD. Any person with ≤ Grade II GVHD must be off systemic immunosuppressive therapy for at least 2 weeks prior to receiving Nivolumab therapy.

2. Inhaled or topical steroids and adrenal replacement steroid doses are permitted in the absence of active auto- or allo-immune disease

3. BMT-related toxicities: patients who developed idiopathic pneumonia syndrome (IPS) or veno-occlusive hepatic disease (VOD) must be off systemic immunosuppression and/or defibrotide for at least 14 days to be eligible.

4. Infection: Patients who have an uncontrolled infection.

5. Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study are not eligible.

6. Has active, known or suspected autoimmune disease. Subjects with vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune thyroiditis only requiring hormone replacement, or conditions not expected to recur in the absence of an external trigger are permitted to enroll.

7. Allergies and Adverse Drug Reaction

   1. History of allergy to study drug components.
   2. History of severe hypersensitivity reaction to any monoclonal antibody.

8. Pregnancy or Breast Feeding: Women of childbearing potential (WOCBP) must agree to follow instructions for method(s) of contraception for the duration of study treatment with nivolumab and 5 months after the last dose of study treatment {i.e., 30 days (duration of ovulatory cycle) plus the time required for the investigational drug to undergo approximately five half-lives. Males who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of study treatment with nivolumab and 7 months after the last dose of study treatment {i.e., 90 days (duration of sperm turnover) plus the time required for the investigational drug to undergo approximately five half-lives.