Inclusion Criteria

          -  ≥2 years and <18 years of age for enrolment in Phase 1 or ≥2 years and ≤21 years of
             age for enrolment in Phase 2.

          -  Recurrent or refractory solid tumors

               -  Phase 1: All solid tumors (measurable or evaluable disease), including primary
                  central nervous system (CNS) tumors; exclusion of hepatoblastoma and lymphomas.
                  Participants with diffuse intrinsic pontine glioma, optic pathway glioma, or
                  pineal tumors with elevated tumor markers (alpha-fetoprotein [AFP] and beta-human
                  chorionic gonadotropin [ß-hCG][or human chorionic gonadotropin [hCG])do not
                  require histological or cytological confirmation of diagnosis

               -  Phase 2: Ewing sarcoma/peripheral primitive neuroectodermal tumor (pPNET),
                  Rhabdomyosarcoma, High Grade Glioma (HGG) (all must have measurable disease);
                  exclusion of Diffuse Intrinsic Pontine Glioma

          -  Histologically or cytologically confirmed diagnosis

          -  Measurable disease that meets the following criteria (Phase 2):

               1. RECIST 1.1 (for all tumor types except HGG): At least 1 lesion of ≥1.0 cm in the
                  longest diameter for a non lymph node or ≥1.5 cm in the short-axis diameter for a
                  lymph node which is serially measurable according to RECIST 1.1 using computed
                  tomography /magnetic resonance imaging (CT/MRI)

               2. Response Assessment in Neuro-Oncology (RANO) for high grade glioma (HGG): At
                  least one lesion must be measurable as defined as a bi dimensionally contrast
                  enhancing lesion with clearly defined margins by CT or MRI scan, with a minimal
                  diameter of 1 cm, and visible on 2 axial slices which are preferably at most 5 mm
                  apart with 0 mm skip

        Lesions that have had external beam radiotherapy (EBRT) or locoregional therapies such as
        radiofrequency (RF) ablation must show evidence of progressive disease based on RECIST 1.1
        to be deemed a target lesion

          -  Karnofsky performance score ≥50 for participants>16 year of age and Lansky play score
             ≥50 for participants ≤16 years of age. Neurologic deficits in participants with CNS
             tumors must have been relatively stable for at least 7 days prior to study enrollment.
             Participants 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

          -  Prior Therapy

               -  Participants must have fully recovered from the acute toxic effects of all prior
                  anti-cancer therapy

               -  Cytotoxic chemotherapy or other chemotherapy known to be myelosuppressive: ≥21
                  days after the last dose of cytotoxic or myelosuppressive chemotherapy (42 days
                  if prior nitrosourea)

               -  Anti-cancer agents not known to be myelosuppressive (eg, not associated with
                  reduced platelet or absolute neutrophil counts): ≥7 days after the last dose of
                  agent

               -  Monoclonal antibodies: ≥21 days or 3 half-lives (whichever is shorter) of the
                  antibody must have elapsed after the last dose of a monoclonal antibody
                  (including checkpoint inhibitors). Toxicity related to prior antibody therapy
                  must be recovered to Grade ≤1

               -  Corticosteroids: If used to modify immune adverse events related to prior
                  therapy, ≥14 days must have elapsed since last dose of corticosteroid.
                  Participants receiving corticosteroids, who have not been on a stable or
                  decreasing dose of corticosteroid for at least 7 days prior to enrollment, are
                  not eligible

               -  Hematopoietic growth factors: ≥14 days after the last dose of a long-acting
                  growth factor or 7 days for short-acting growth factor. For agents that have
                  known adverse events occurring beyond 7 days after administration, this period
                  must be extended beyond the time during which adverse events are known to occur

               -  Interleukins, interferons, and cytokines (other than hematopoietic growth
                  factors): ≥21 days after the completion of interleukins, interferons or cytokines
                  (other than hematopoietic growth factors)

               -  Stem cell infusions (with or without total body irradiation): Allogeneic
                  (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion
                  including donor leukocytes infusion or boost infusion: ≥84 days after infusion
                  and no evidence of graft versus host disease; Autologous stem cell infusion
                  including boost infusion: ≥42 days

               -  Cellular Therapy: ≥42 days after the completion of any type of cellular therapy
                  (eg, modified T cells, natural killer cells, dendritic cells, etc)

               -  Radiotherapy (XRT)/External Beam Irradiation including Protons: ≥14 days after
                  local XRT; ≥150 days after total body irradiation, craniospinal XRT or if
                  radiation to ≥50% of the pelvis; ≥42 days if other substantial bone marrow
                  radiation.

               -  Radiopharmaceutical therapy: ≥42 days after systemically administered therapy.

               -  Vascular endothelial growth factor (VEGF)/VEGF receptor (VEGFR)-targeted or
                  mammalian target of rapamycin (mTOR)-targeted therapies: Must not have received
                  prior exposure to lenvatinib; May have previously progressed on an mTOR
                  inhibitor; No more than 2 prior VEGF/VEGFR-targeted therapies (For Phase 2 only);
                  Must not have received prior VEGF/VEGFR-targeted therapy in combination with an
                  mTOR inhibitor (For Phase 2 only)

          -  Adequate bone marrow function for participants with solid tumors without known bone
             marrow involvement

          -  Adequate bone marrow function for participants with known bone marrow metastatic
             disease

          -  Adequate renal function

          -  Adequate liver function

          -  Adequate cardiac function

          -  Adequate neurologic function

          -  Adequate blood pressure (BP) control with or without antihypertensive medications

          -  Adequate coagulation

          -  Adequate pancreatic function

          -  Adequate metabolic function

          -  Adequate glycemic control

          -  Participants must have a minimum body surface area (BSA) of 0.6 m^2 at study entry.

        Exclusion Criteria

          -  Participants who have had or are planning to have the following invasive procedures

               -  Major surgical procedure, laparoscopic procedure, open biopsy or significant
                  traumatic injury within 28 days prior to enrolment

               -  Central line placement or subcutaneous port placement is not considered major
                  surgery. External central lines must be placed at least 3 days prior to
                  enrollment and subcutaneous ports must be placed at least 7 days prior to
                  enrollment

               -  Fine needle aspirate within 7 days prior to enrolment

               -  Surgical or other wounds must be adequately healed prior to enrolment

               -  For purposes of this study, bone marrow aspirate and biopsy are not considered
                  surgical procedures and therefore are permitted within 14 days prior to start of
                  protocol therapy

          -  Participants who have non-healing wound, unhealed or incompletely healed fracture, or
             a compound (open) bone fracture at the time of enrolment

          -  Participants having an active infection requiring systemic therapy.

          -  Participants with a known history of active hepatitis B (defined as hepatitis B
             surface antigen reactive or hepatitis B virus- deoxyribonucleic [DNA] detected) or
             known active hepatitis C virus (HCV, defined as HCV- Ribonucleic acid [RNA] detected).
             Note: No testing for hepatitis B and hepatitis C is required unless mandated by the
             local health authority.

          -  Known to be human immunodeficiency virus (HIV) positive. Note: HIV testing is required
             at screening only when mandated by the local health authority

          -  Clinical evidence of nephrotic syndrome prior to enrolment

          -  Gastrointestinal bleeding or active hemoptysis (bright red blood of at least half
             teaspoon) within 21 days prior to enrolment

          -  Thrombotic/ thromboembolic event requiring systemic anticoagulation within 90 days
             prior to enrollment

          -  Evidence of new intracranial hemorrhage of more than punctate size on MRI assessment
             obtained within 28 days prior to study enrollment for Participants with HGG

          -  Diagnosis of lymphoma

          -  Radiographic evidence of major blood vessel invasion/infiltration.

          -  Evidence of untreated CNS metastases (exception: participants with primary CNS tumors
             and leptomeningeal disease)

          -  Participants who are currently receiving enzyme-inducing anticonvulsants

          -  Participants chronically receiving strong cytochrome P450 3A4 (CYP3A4)/P-glycoprotein
             (P-gp) inhibitors or inducers within 7 days prior to study enrollment

          -  Females who are breastfeeding or pregnant. For females of childbearing potential, a
             negative screening pregnancy test must be obtained within 72 hours before the first
             dose of study drug