Inclusion Criteria:

          -  Patient must have enrolled onto APEC1621SC and must have been given a treatment
             assignment to molecular analysis for therapy choice (MATCH) to APEC1621E based on the
             presence of an actionable mutation

               -  Note: patients with BRAF V600 actionable mutations of interest (aMOIs) will be
                  preferentially assigned to APEC1621G (vemurafenib) if that study is open and they
                  are otherwise eligible for it

          -  Patients must have a body surface area >= 0.5 m^2 at enrollment

          -  Patients must have radiographically measurable disease at the time of study
             enrollment; patients with neuroblastoma who do not have measurable disease but have
             iobenguane (MIBG) positive (+) evaluable disease are eligible; measurable disease in
             patients with CNS involvement is defined as tumor that is measurable in two
             perpendicular diameters on magnetic resonance imaging (MRI) and visible on more than
             one slice

               -  Note: the following do not qualify as measurable disease:

                    -  Malignant fluid collections (e.g., ascites, pleural effusions)

                    -  Bone marrow infiltration except that detected by MIBG scan for neuroblastoma

                    -  Lesions only detected by nuclear medicine studies (e.g., bone, gallium or
                       positron emission tomography [PET] scans) except as noted for neuroblastoma

                    -  Elevated tumor markers in plasma or cerebrospinal fluid (CSF)

                    -  Previously radiated lesions that have not demonstrated clear progression
                       post radiation

                    -  Leptomeningeal lesions that do not meet the measurement requirements for
                       Response Evaluation Criteria in Solid Tumors (RECIST) 1.1

          -  Karnofsky >= 50% for patients > 16 years of age and Lansky >= 50 for patients =< 16
             years of age; Note: neurologic deficits in patients with CNS tumors must have been
             stable for at least 7 days prior to study enrollment; 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 must have fully recovered from the acute toxic effects of all prior
             anti-cancer therapy and must meet the following minimum duration from prior
             anti-cancer directed therapy prior to enrollment; if after the required timeframe, the
             numerical eligibility criteria are met, e.g. blood count criteria, the patient is
             considered to have recovered adequately

               -  Cytotoxic chemotherapy or other anti-cancer agents 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 (e.g. not associated with
                  reduced platelet or absolute neutrophil count [ANC] counts): >= 7 days after the
                  last dose of agent

               -  Antibodies: >= 21 days must have elapsed from infusion of last dose of antibody,
                  and 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

               -  Hematopoietic growth factors: >= 14 days after the last dose of a long-acting
                  growth factor (e.g. pegfilgrastim) or 7 days for short-acting growth factor; for
                  growth factors 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; the duration of this interval must be discussed with
                  the study chair and the study-assigned research coordinator

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

               -  Stem cell Infusions (with or without total body irradiation [TBI]):

                    -  Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem
                       cell infusion including donor lymphocyte infusion (DLI) or boost infusion:
                       >= 84 days after infusion and no evidence of graft versus host disease
                       (GVHD)

                    -  Autologous stem cell infusion including boost infusion: >= 42 days

               -  Cellular therapy: >= 42 days after the completion of any type of cellular therapy
                  (e.g. modified T cells, natural killer [NK] cells, dendritic cells, etc.)

               -  X-ray therapy (XRT)/external beam irradiation including protons: >= 14 days after
                  local XRT; >= 150 days after TBI, craniospinal XRT or if radiation to >= 50% of
                  the pelvis; >= 42 days if other substantial bone marrow (BM) radiation

                    -  Note: radiation may not be delivered to "measurable disease" tumor site(s)
                       being used to follow response to subprotocol treatment

               -  Radiopharmaceutical therapy (e.g., radiolabeled antibody, iodine I 131
                  metaiodobenzylguanidine [131I-MIBG]): >= 42 days after systemically administered
                  radiopharmaceutical therapy

               -  Patients must not have received prior exposure to selumetinib (AZD6244 hydrogen
                  sulfate)

          -  For patients with solid tumors without known bone marrow involvement:

               -  Peripheral absolute neutrophil count (ANC) >= 1000/mm^3 within 7 days prior to
                  enrollment

               -  Platelet count >= 100,000/mm^3 (transfusion independent, defined as not receiving
                  platelet transfusions for at least 7 days prior to enrollment)

          -  Patients with known bone marrow metastatic disease will be eligible for study provided
             they meet the blood counts (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

          -  Creatinine clearance or radioisotope glomerular filtration rate (GFR) within 7 days
             prior to enrollment >= 70 ml/min/1.73 m^2 or a serum creatinine based on age/gender as
             follows:

               -  Age: 1 to < 2 years; maximum serum creatinine (mg/dL): male 0.6; female 0.6

               -  Age: 2 to < 6 years; maximum serum creatinine (mg/dL): male 0.8; female 0.8

               -  Age: 6 to < 10 years; maximum serum creatinine (mg/dL): male 1; female 1

               -  Age: 10 to < 13 years; maximum serum creatinine (mg/dL): male 1.2; female 1.2

               -  Age: 13 to < 16 years; maximum serum creatinine (mg/dL): male 1.5; female 1.4

               -  Age: >= 16 years; maximum serum creatinine (mg/dL): male 1.7; female 1.4

          -  Bilirubin (sum of conjugated + unconjugated) =< 1.5 x upper limit of normal (ULN) for
             age within 7 days prior to enrollment

          -  Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase [ALT]) =< 135
             U/L; (for the purpose of this study, the ULN for SGPT is 45 U/L) within 7 days prior
             to enrollment

          -  Serum albumin >= 2 g/dL within 7 days prior to enrollment

          -  Shortening fraction of >= 27% by echocardiogram, or ejection fraction of >= 50% by
             gated radionuclide study within 7 days prior to enrollment

          -  A blood pressure (BP) =< the 95th percentile for age, height, and gender measured
             within 7 days prior to enrollment; please note that 3 serial blood pressures should be
             obtained and averaged to determine baseline BP; patients with hypertension controlled
             on antihypertensive medications will be allowed if otherwise eligible

          -  Serum triglyceride level =< 300 mg/dL within 7 days prior to enrollment

          -  Serum total cholesterol level =< 300 mg/dL within 7 days prior to enrollment

          -  Patients must be able to swallow intact capsules whole

          -  All patients and/or their parents or legally authorized representatives must sign a
             written informed consent; assent, when appropriate, will be obtained according to
             institutional guidelines

        Exclusion Criteria:

          -  Pregnant or breast-feeding women will not be entered on this study because there is
             currently no available information regarding human fetal or teratogenic toxicities;
             pregnancy tests must be obtained in girls who are post-menarchal; females of
             reproductive potential may not participate unless they have agreed to use an effective
             contraceptive method for the duration of study treatment; males with sexual partners
             who are pregnant or who could become pregnant (ie, women of child-bearing potential)
             should use effective methods of contraception for 12 weeks after completing the study
             to avoid pregnancy and/or potential adverse effects on the developing embryo

          -  Concomitant medications

               -  Corticosteroids: patients 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; if used to modify immune adverse events related to prior therapy,
                  >= 14 days must have elapsed since last dose of corticosteroid

               -  Investigational drugs: patients who are currently receiving another
                  investigational drug are not eligible

               -  Anti-cancer agents: patients who are currently receiving other anti-cancer agents
                  are not eligible

               -  Anti-GVHD agents post-transplant: patients who are receiving cyclosporine,
                  tacrolimus or other agents to prevent graft-versus-host disease post bone marrow
                  transplant are not eligible for this trial

               -  CYP3A4 agents: patients who are currently receiving drugs that are strong
                  inducers or inhibitors of CYP3A4 are not eligible; strong inducers or inhibitors
                  of CYP3A4 should be avoided from 14 days prior to enrollment to the end of the
                  study

               -  CYP2C19 agents: patients who are currently receiving drugs that are strong
                  CYP2C19 inducers (e.g., rifampin, ritonavir) or inhibitors (e.g.., fluoxetine,
                  fluvoxamine, ticlopidine) are not eligible

          -  Patients who have an uncontrolled infection are not eligible

          -  Patients with known significant ophthalmologic conditions (uncontrolled glaucoma,
             history of retinal vein occlusion or retinal detachment, excluding patients with
             longstanding findings secondary to existing conditions) are not eligible

          -  Patients with low grade glioma are not eligible

          -  Patients who have received a prior solid organ transplantation are not eligible

          -  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