Inclusion Criteria:

          -  Diagnosis:

               -  Phase I cohort: Adults >= 18 years with relapsed/refractory FLT3-mutated AML or
                  myelodysplastic syndrome (MDS) that is intermediate-2 or high-risk by the
                  International Prognostic Scoring System

               -  Phase II cohort A: Adults >= 18 years with newly diagnosed FLT3-mutated AML.
                  Patients should meet the following criteria:

                    -  Confirmed newly diagnosed AML with FLT3 mutation

                    -  Ineligible for induction therapy defined as

                         -  Either age >= 75

                         -  Or 18-74 with at least one comorbidity (congestive heart failure [CHF]
                            requiring therapy or ejection fraction [EF] =< 50%, diffusion capacity
                            of the lung for carbon monoxide [DLCO] =< 65% or forced expiratory
                            volume in 1 second [FEV1] =< 65%, or Eastern Cooperative Oncology Group
                            [ECOG] 2 or 3, or other significant co-morbidity precluding use of
                            cytotoxic chemotherapy as approved by the principal investigator (PI)

               -  Phase II cohort B: Adults >= 18 years with relapsed/refractory FLT3-mutated AML
                  or MDS that is intermediate-2 or high-risk by the International Prognostic
                  Scoring System who have received 1 prior therapy

               -  For all cohorts, patients with either FLT3-ITD or FLT3 D835 mutations will be
                  eligible

          -  Performance status =< 3 (Eastern Cooperative Oncology Group [ECOG] scale)

          -  Total serum bilirubin =< 2.5 x upper limit of normal (ULN), unless due to Gilbert's
             syndrome, hemolysis or the underlying leukemia approved by the PI

          -  Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) =< 3 x ULN, unless
             due to the underlying leukemia approved by the PI

          -  Creatinine clearance >= 30 mL/min

          -  Ability to swallow

          -  Signed informed consent

          -  Hydroxyurea or one dose of cytarabine up to 1000 mg is allowed to reduce the white
             blood cell (WBC) to less than 25 x 10^9/L prior to initiation of study treatment

        Exclusion Criteria:

          -  Prior therapies

               -  Phase I cohort: No restriction based on prior therapies

               -  Phase II cohort A: Patients with prior therapy for AML are not eligible. Prior
                  therapy for antecedent hematologic disorder is allowed including prior
                  hypomethylating agent (HMA) therapy for MDS. Prior hydroxyurea or cytarabine
                  given for purposes of cytoreduction is also allowed. Prior all trans-retinoic
                  acid given for presumed acute promyelocytic leukemia is also allowed

               -  Phase II cohort B: Patients with >= 3 prior lines of therapy are not eligible.
                  Stem cell transplantation, treatment given only for cytoreductive purposes (e.g.
                  hydroxyurea), and growth factors do not count as lines of therapy for this
                  purpose. Prior therapy with venetoclax is allowed

          -  Prior treatment with gilteritinib

          -  Patients suitable for and willing to receive intensive induction chemotherapy (for
             Phase II cohort A only)

          -  Congenital long QT syndrome or corrected QT (QTc) > 450 msec. Repeat
             electrocardiograms (EKGs) after correction of electrolytes or discontinuation of QT
             prolonging medications are allowed to meet entry criteria

          -  Active serious infection not controlled by oral or intravenous antibiotics (e.g.
             persistent fever or lack of improvement despite antimicrobial treatment)

          -  Active grade III-V cardiac failure as defined by the New York Heart Association
             Criteria

          -  Active central nervous system leukemia

          -  Known history of human immunodeficiency virus (HIV) seropositive

          -  Known hepatitis B surface antigen seropositive or known or suspected active hepatitis
             C infection

               -  Note: Patients who have isolated positive hepatitis B core antibody (i.e., in the
                  setting of negative hepatitis B surface antigen and negative hepatitis B surface
                  antibody) must have an undetectable hepatitis B viral load. Patients who have
                  positive hepatitis C antibody may be included if they have an undetectable
                  hepatitis C viral load

          -  Patients with a prior or concurrent malignancy whose natural history or treatment is
             not anticipated to interfere with the safety or efficacy assessment of the
             investigational regimen may be included only after discussion with the PI

          -  Consumed strong inducer of CYP3A or p-glycoprotein within 3 days of study enrollment.
             Agents include but are not limited to: carbamazepine, phenytoin, rifampin, and St.
             John's wart

          -  Treatment with any investigational antileukemic agents or chemotherapy agents in the
             last 7 days before study entry, unless full recovery from side effects has occurred or
             patient has rapidly progressive disease judged to be life-threatening by the
             investigator. Prior recent treatment with corticosteroids, hydroxyurea and/or
             cytarabine (given for cytoreduction) permitted. Use of hydroxyurea or one dose
             cytarabine to reduce WBC below 25 prior to initiation of study treatment is
             recommended

          -  Pregnant women will not be eligible; women of childbearing potential should have a
             negative pregnancy test prior to entering on the study and be willing to use effective
             methods of contraception throughout the study period and for at least 6 months after
             the last dose of study drugs. Women do not have childbearing potential if they have
             had a hysterectomy or are postmenopausal without menses for 12 months. In addition,
             men enrolled on this study should understand the risks to any sexual partner of
             childbearing potential and should practice an effective method of birth control
             throughout the study period and for at least 4 months after the last dose of study
             drugs. Lactating women (or those planning to breastfeed) should not breastfeed during
             treatment of gilteritinib and for at least 2 months after the last dose of
             gilteritinib