Inclusion Criteria:

          -  Subject is aged ≥ 6 months and < 21 years of age* at the time of signing informed
             consent and/or assent, as applicable.

               -  *For phase 2: Enrollment of subjects from 6 months to less than 1 year and 1 year
                  to less than 2 years will be dependent on the establishment of recommended phase
                  2 dose (RP2D) in the respective age groups during phase 1.

          -  Subject has a diagnosis of acute myeloid leukemia (AML) according to The
             French-American-British (FAB) classification with ≥ 5% blasts in the bone marrow, with
             or without extramedullary disease (except subjects with active central nervous system
             [CNS] leukemia).

               -  In the phase 1 portion of the study, subject must be in first or greater relapse
                  or refractory to induction therapy with no more than 1 attempt at remission
                  induction.

               -  For the phase 2 portion of the study, subject must be in first relapse.

          -  Subject has fully recovered from the acute toxic effects of all prior chemotherapy,
             immunotherapy, or radiotherapy prior to entering this study.

               -  Myelosuppressive chemotherapy:

                    -  For subject who relapses while receiving cytotoxic therapy, at least 21 days
                       must have elapsed since the completion of cytotoxic therapy and prior to
                       screening.

                    -  Cytoreduction with hydroxyurea can be initiated and continued for up to 24
                       hours prior to the start of systemic protocol therapy (cycle 1 day -1).
                       Subject may also receive low dose cytarabine (100 mg/m^2 per dose once daily
                       for 5 days) for cytoreduction until 24 hours prior to cycle 1 day -1.

                    -  Subject who has received other FLT3 inhibitors (e.g., lestaurtinib,
                       sorafenib, etc) is eligible for this study.

               -  Hematopoietic growth factors: at least 7 days must have elapsed since the
                  completion of therapy with a growth factor and prior to screening.

               -  Biologic (anti-neoplastic agent): at least 7 days must have elapsed since the
                  completion of therapy with a biologic agent and prior to screening. For agents
                  that have known adverse events (AEs) occurring beyond 7 days after
                  administration, this period must be extended beyond the time during which AEs are
                  known to occur.

               -  X-ray treatment (XRT):

                    -  14 days must have elapsed for local palliative XRT for CNS chloromas and
                       prior to screening; no washout period is necessary for other chloromas;

                    -  Prior to screening, 90 days must have elapsed if the subject had a prior
                       traumatic brain injury or has received craniospinal XRT.

          -  For subject undergoing hematopoietic stem cell transplant (HSCT), at least 90 days
             must have elapsed since HSCT and subject must not have active graft-versus-host
             disease (GVHD).

          -  Subject has Karnofsky score ≥ 50 (if the subject is of ≥ 16 years of age) or Lansky
             score of ≥ 50 (if the subject is < 16 years of age). A score < 50 is acceptable if
             related to the subject's leukemia.

          -  Subject must meet the following criteria as indicated on the clinical laboratory
             tests.

               -  Serum aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3.0 x
                  upper limit normal (ULN) for age

               -  Total serum bilirubin ≤ 1.5 x ULN for age

               -  Serum creatinine ≤ 1.5 x ULN for age or an estimated glomerular filtration rate
                  of > 60 mL/min/1.73 m^2.

          -  A female subject is eligible to participate if she is not pregnant and at least 1 of
             the following conditions applies:

               -  Not a woman of childbearing potential (WOCBP) OR

               -  WOCBP who agrees to follow the contraceptive guidance throughout the treatment
                  period and for at least 180 days after the final study drug administration.

          -  Female subject must agree not to breastfeed starting at Screening, and throughout the
             study period and for 60 days after the final study drug administration.

          -  Female subject must not donate ova starting at Screening and throughout the study, and
             for 180 days after the final study drug administration.

          -  A male subject with female partner(s) of childbearing potential must agree to use
             contraception during the treatment period and for at least 180 days after the final
             study drug administration.

          -  A male subject must not donate sperm during the treatment period and for at least 120
             days after the final study drug administration.

          -  Male subject with a pregnant or breastfeeding partner(s) must agree to remain
             abstinent or use a condom for the duration of the pregnancy or time partner is
             breastfeeding throughout the study period and for 180 days after the final study drug
             administration.

          -  Subject and subject's parent(s) or legal guardian agrees not to participate in another
             interventional study while on treatment.

          -  Live Vaccines - At least 6 weeks must have elapsed since the administration of the
             last dose of a live vaccine and prior to the initiation of study treatment (cycle 1,
             day -1)

          -  Phase 1: Subject is positive for FLT3 (ITD and/or tyrosine kinase domain [TKD])
             mutation in bone marrow or blood as determined by the local institution except for
             sites in the USA. USA sites - subject is positive for the FLT3 (ITD) mutation.

          -  Phase 2: Subject is positive for the FLT3 (ITD) mutation in bone marrow or blood as
             determined by the local institution.

        Exclusion Criteria:

          -  Subject has active CNS leukemia.

          -  Subject has uncontrolled or significant cardiovascular disease, including:

               -  Diagnosed or suspected congenital long QT syndrome or any history of clinically
                  significant ventricular arrhythmias (such as ventricular tachycardia, ventricular
                  fibrillation, or Torsades de Pointes (TdP)); any history of arrhythmia will be
                  discussed with the sponsor prior to subject's entry into the study

               -  Prolonged Fridericia's Correction Formula (QTcF) interval on pre-entry
                  electrocardiogram (ECG) (≥ 450 ms)

               -  Any history of second or third degree heart block (may be eligible if the subject
                  currently has a pacemaker)

               -  Heart rate < 50 beats/minute on pre-entry ECG

               -  Uncontrolled hypertension

               -  Complete left bundle branch block

          -  Subject has systemic fungal, bacterial, viral or other infection that is exhibiting
             ongoing signs/symptoms related to the infection without improvement despite
             appropriate antibiotics or other treatment. The subject needs to be off pressors and
             have negative blood cultures for 48 hours.

          -  Subject is receiving or plans to receive concomitant chemotherapy, radiation therapy,
             or immunotherapy other than as specified in the protocol.

          -  Subject has active clinically significant graft-versus-host disease (GVHD) or is on
             treatment with systemic corticosteroids and is receiving > 0.5 mg/kg of prednisone (or
             equivalent) daily dose for GVHD.

          -  Subject has active malignant tumors other than AML.

          -  Subject has any significant concurrent disease, illness, psychiatric disorder or
             social issue that would compromise subject safety or compliance; interfere with
             consent, study participation, follow-up or interpretation of study results.

          -  Subject has hypokalemia and/or hypomagnesemia at Screening (defined as values below
             institutional lower limit of normal [LLN]). Repletion of potassium and magnesium
             levels during the screening period is allowed.

          -  Subject requires treatment with concomitant drugs that are strong inducers of
             cytochrome P450 (CYP)3A/P-glycoprotein (P-gp).

          -  Subject is known to have human immunodeficiency virus infection.

          -  Subject has active hepatitis B or C, or other active hepatic disorder.

               -  Subjects with positive hepatitis B surface antigen (HBsAg) or detectable
                  hepatitis B DNA are not eligible.

               -  Subjects with negative HBsAg, positive hepatitis B core antibody and negative
                  hepatitis B surface antibody will be eligible if hepatitis B DNA is undetectable.

               -  Subjects with antibodies to hepatitis C virus will be eligible if hepatitis C RNA
                  is undetectable.

          -  Subject must wait for at least 5 half-lives after stopping therapy with any
             investigational agent and before starting gilteritinib.

          -  Subject has a known or suspected hypersensitivity to gilteritinib, cytarabine,
             fludarabine, granulocyte colony-stimulating factor (G-CSF) or any components of the
             formulation used.