Inclusion Criteria:  Patients with each of the below pediatric tumor types, having achieved a "best response" (BR) defined as:  Absence of measurable disease by CT or MRI 4 weeks off any antineoplastic therapy OR Absence of evaluable disease by 18-FDG PET-CT or PET-MRI on two scans at least 4 weeks apart, while off therapy, despite the presence of measurable disease by conventional CT or MRI (e.g., after irradiation of target lesions) OR Stable measurable disease with <10% change in size of any target lesion, as measured by CT or MRI, after 4 weeks without any antineoplastic therapy, independent of MIBG or PET avidity  Stratum 1: Neuroblastoma (target enrollment 35 patients)  Neuroblastoma in best response 2 (i.e., re-induced to best response after prior relapse) or greater (BR2+); this includes patients who had refractory disease or progressed during upfront therapy and attained a stable best response with salvage therapy; this does NOT include patients who received additional treatment for partial response to induction therapy; specific cases may be discussed with the study chair for clarification Neuroblastoma in BR1 in patient >5 years of age and stage M at diagnosis without a complete response at the end of induction therapy Neuroblastoma in BR1 at the end of frontline therapy but with residual stable disease (including MIBG-avid stable disease as per section 3.1.1 or >5% bone marrow involvement) Patients with neuroblastoma must also have stable (≤20% variance) or falling urine HVA/Cr and VMA/Cr over two labs obtained 4 weeks apart  Stratum 2: Central nervous system (CNS) tumors (target enrollment 29 patients)  Atypical teratoid rhabdoid tumors (ATRT), BR1+ Diffuse intrinsic pontine glioma (DIPG), BR1+ having completed upfront radiation therapy High grade gliomas (HGG, WHO Grade 3 or 4 astrocytomas) in BR1+ Medulloblastoma, Molecular Group 3 (with MYC/MYCN expression/amplification) or Group 4 in BR1+ Medulloblastoma in any molecular group in BR2+ Germ Cell tumor of the CNS, BR2+  Stratum 3: Sarcomas and other solid tumors (target enrollment 36 patients)  Malignant rhabdoid Tumor, BR1+ Osteosarcoma, any BR2+ Ewing sarcoma with metastases not undergoing metastatectomy, BR1+ Ewing sarcoma, any BR2+ Rhabdomyosarcoma, BR1 with irradiated positive margins, Rhabdomyosarcoma, BR2+, alveolar subtype or fusion-positive subtype Rhabdomyosarcoma, BR2+, embryonal subtype, Group 4 at original diagnosis Desmoplastic small round blue cell tumor, BR1+ Any other soft tissue sarcoma, BR2+ Germ cell tumors outside the CNS, BR2+ Wilms tumor, diffuse anaplasia histology, any stage, BR2+ Wilms tumor, relapse <12 months, prior treatment with doxorubicin, and intraabdominal recurrence, any histology, BR2+ Other solid tumors in which patients had refractory or progressive disease to initial therapy, excluding cases of surgery alone, and were then able to attain a stable best response with a salvage regimen Other solid tumors not arising from the head with <25% 1-year EFS, per discussion with the principal investigator  Other Inclusion Criteria:  Age: ≥ 18 months of age and <40 years of age at time of study entry Performance level: Patients must have a Lansky or Karnofsky performance status score of ≥ 50, corresponding to ECOG categories 0, 1 or 2. Use Karnofsky for patients > 16 years of age and Lansky for patients ≤ 16 years of age. Patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. Patients should also have recovery to baseline or ≤ Grade 1 CTCAE v4.03 from toxicities related to any prior treatments, unless AE(s) are clinically nonsignificant and/or stable on supportive therapy (as further clarified below) Patient Body Surface Area (BSA): Patients must be ≥0.35 m2 in BSA at time of diagnosis  Prior therapy: patients must have recovered from the acute toxic effects of prior therapy, with the following time specifications:  Myelosuppressive chemotherapy: Patients must not have received myelosuppressive chemotherapy within 3 weeks of enrollment on study (6 weeks if prior therapy included nitrosourea) Other medicinal anti-cancer agents: Patients must not have received nonmyelosuppressive anticancer agents, including any type of small molecule kinase inhibitor, within 14 days of enrollment on study Biological anticancer therapy (including antibody therapy) Patients must not have received biological anticancer therapy within 21 days of enrollment on study Radiation therapy: Patients (except in stratum 4) must not have received external beam radiation therapy to sites outside of the lungs within 2 weeks of study enrollment, external beam radiation therapy to sites within the lungs within 4 weeks of study enrollment, or MIBG therapy within 6 weeks of study enrollment. Subjects with clinically relevant ongoing complications from prior radiation therapy are not eligible. Myeloablative therapy: Patients must not have received myeloablative therapy within 2 months of study enrollment and must not have received a blood stem cell/marrow infusion within 3 weeks of study enrollment and must have attained blood count recovery as defined below. Patient must not have been on agents for control of graft versus host disease for at least 4 weeks prior to study enrollment Bone Marrow Function: Patients must have adequate bone marrow function at time of study enrollment, as defined as: Absolute neutrophil count (ANC) ≥1000/mcL; patients cannot have received filgrastim, pegfilgrastim or equivalent biosimilar within 14 days of study enrollment Platelet count ≥ 100,000/mcL; patients can receive no more than 15 mL/kg of platelet transfusions per week at time of enrollment to meet the parameters; patients can receive a TPO agonist (e.g., eltrombopag or romiplostim) at time of enrollment but must be on a stable dose for at least 14 days prior to enrollment Hemoglobin ≥ 8.0 g/dL; patients can receive no more than 10 mL/kg of packed red blood cells (PRBCs)/week transfused at time of enrollment on therapy to meet the parameters; patients may receive erythropoietin or biosimilar equivalent but must have been on a stable dose and not require PRBC transfusions for at least 14 days prior to study enrollment Patients should also have <5% bone marrow involvement by tumor if bone marrows have been evaluated as part of clinical standard of care Renal Function: Patients must have a creatinine clearance, as defined by the modified Schwartz formula or by radioisotope GFR, of ≥70 mL/min/1.72 m2, and must have urine protein ≤ 30 mg/dL or ≤+1 of dipstick or quantitative protein <1000 mg in a 24 hr urine sample.  Hepatic function: patients must have adequate hepatic function, defined as:  total bilirubin <2x upper limit of normal for age ALT<5x ULN Serum albumin >2.7 g/dL  Cardiovascular Function: Patients must have adequate cardiovascular function as defined as:  No significant arrhythmias, strokes, or myocardial infarction within 6 months of study enrollment QTc ≤ 480 msec at time of study enrollment Blood pressure ≤ 95th percentile for age, height, and gender for patients <18 years of age (78), or BP ≤140/90 for patients ≥18 years of age. At time of enrollment, patients may be on one antihypertensive agent at a stable dose for at least 4 weeks prior to enrollment. Pancreatic function: Patient must have adequate pancreatic function, as defined by a serum lipase <2x ULN Neurologic function: Patients with defined seizures who are on a stable anti- convulsant regimen using drugs that do not induce hepatic metabolizing enzymes for at least 4 weeks are eligible for enrollment. Lung integrity: Patients must not have had any invasive pulmonary procedure (including bronchoalveolar lavage, lung biopsy, transbronchial biopsy, or thoracotomy) or pneumothorax within 4 weeks of enrollment on study. Surgeries or trauma: Patients must not have had any major surgical procedure, laparoscopic procedure, sepsis, shock, or trauma within 4 weeks of the start of therapy. Patients must not have had a central line or subcutaneous port placement, revision, or removal (excluding a peripherally inserted central catheter (PICC)) within 7 days of the start of therapy. Patients must not have had a core or fine needle biopsy within 7 days of the start of therapy. The primary surgeon of any major surgical procedures must authorize antineoplastic treatment before enrollment on study. Any wounds or incisions must be healed prior to enrollment on study. Bone marrow aspiration and/or biopsy are not considered surgical procedures for the purpose of this study. Patients must be able to swallow tablets intact. Tablets cannot be cut or crushed. Sexually active fertile subjects and their partners must agree to use medically accepted methods of contraception (eg, barrier methods, including male condom, female condom, or diaphragm with spermicidal gel) during the course of the study and for 4 months after the last dose of study treatment. Patient or legal guardian must be capable of understanding and complying with the protocol requirements and must have signed the informed consent document.  Exclusion Criteria:  Prior treatment with cabozantinib Women who are pregnant or breastfeeding will not be enrolled on this study due to potential teratogenic or development toxicities. Post-menarchal females must be confirmed to not be pregnant at time of enrollment and each month of treatment. Males and females of reproductive age and potential must agree to two forms of birth control between themselves and their partner(s) throughout the treatment period and for four months after the final dose of cabozantinib Patients requiring corticosteroids who have not been on a stable or decreasing dose of corticosteroid for the 7 days prior to enrollment are not eligible. If used to modify immune adverse events related to prior therapy, ≥ 14 days must have elapsed since last dose of corticosteroid. Patients may not be on other chronic immunosuppressive agents for at least 14 days prior to enrollment on study. Patients with active graft versus host disease Patients on other anticancer agents, either as experimental therapies, standard of care or off-label are not eligible for enrollment. Patients must not have taken a strong CYP3A4 inhibitor or inducer within 14 days of the first dose of cabozantinib. It is encouraged for patients taking other inducers or inhibitors of CYP3A4 be changed to another appropriate drug during the period of cabozantinib administration. Patients on anticoagulation treatment are not eligible for enrollment. Patients on anticoagulation prophylaxis for a history of thrombosis (diagnosed >6 weeks prior and no longer on treatment dosing) can remain on anticoagulation during study with standard of care agents, but not on experimental agents. Patients cannot receive betrixaban or dabigatran for anticoagulation. The subject has prothrombin time (PT)/INR or partial thromboplastin time (PTT) test ≥ 1.3 x the laboratory ULN within 7 days before the first dose of study treatment. Patients with an uncontrolled infection are ineligible for enrollment. Patients who, in the opinion of the investigator, are not able to comply with safety monitoring requirements are not eligible for enrollment. Patients with radiation-related mucocutaneous injury, either primary or related to radiation recall or false photosensitivity, are not eligible for enrollment until toxicities have resolved for at least 7 days.  Cardiovascular disorders:  Congestive heart failure New York Heart Association Class 3 or 4, unstable angina pectoris, serious cardiac arrhythmias. Uncontrolled hypertension defined as sustained blood pressure (BP) > 140 mm Hg systolic or > 90 mm Hg diastolic despite optimal antihypertensive treatment. Stroke (including transient ischemic attack [TIA]), myocardial infarction (MI), or other ischemic event, or thromboembolic event (eg, deep venous thrombosis, pulmonary embolism) within 6 months before first dose, excluding uncomplicated central venous catheter-associated thrombus that is not expanding, with or without stable use of anticoagulation. Gastrointestinal (GI) disorders including those associated with a high risk of perforation or fistula formation: The subject has evidence of tumor invading the GI tract, active peptic ulcer disease, inflammatory bowel disease (eg, Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, acute pancreatitis, acute obstruction of the pancreatic duct or common bile duct, or gastric outlet obstruction. Abdominal fistula, GI perforation, bowel obstruction, or intra-abdominal abscess within 6 months before first dose. Clinically significant hematuria, hematemesis, or hemoptysis of > 0.5 teaspoon (2.5 ml) of red blood, or other history of significant bleeding (eg, pulmonary hemorrhage) within 12 weeks before first dose. Cavitating pulmonary lesion(s) or known endotracheal or endobronchial disease manifestation. Lesions invading or encasing any major blood vessels. Other clinically significant disorders that would preclude safe study participation. Serious non-healing wound/ulcer/bone fracture. Uncompensated/symptomatic hypothyroidism; patients with controlled hypothyroidism with the use of thyroid replacement therapy are eligible for inclusion. Moderate to severe hepatic impairment (Child-Pugh B or C). Major surgery (eg, GI surgery, removal or biopsy of brain metastasis) within 8 weeks before first dose of study treatment. Complete wound healing from major surgery must have occurred 1 month before first dose and from minor surgery (eg, simple excision, tooth extraction) at least 10 days before first dose. Subjects with clinically relevant ongoing complications from prior surgery are not eligible. Corrected QT interval calculated by the Fridericia formula (QTcF) > 480 ms per electrocardiogram (ECG) within 28 days before first dose of study treatment [add reference for Fridericia formula]. Previously identified allergy or hypersensitivity to components of the study treatment formulations. Diagnosis of another malignancy within 2 years before first dose of study treatment, except for superficial skin cancers, or localized, low grade tumors deemed cured and not treated with systemic therapy.