Inclusion Criteria:

          -  Diagnosis of

               -  AML (World Health Organization [WHO] classification definition of >= 20% blasts,
                  excluding Acute promyelocytic leukemia),

               -  Acute biphenotypic leukemia or

               -  High-risk MDS (> 10% bone marrow blasts)

          -  Frontline cohort: Patients aged 18 to 65 years

          -  Relapse cohort: Patients aged >=18 years old

          -  Patients may be newly diagnosed (Frontline cohort) or with prior therapy (Relapsed
             cohort) as follows:

               -  For frontline cohort: Patients must be chemonaive, i.e., not have received any
                  chemotherapy (except hydroxyurea [Hydrea] [no dose limit], tretinoin [atra] [no
                  dose limit] or ara-C [one or two doses (max 2 gr/m^2 per dose)] for transient
                  control of hyperleukocytosis) for AML or MDS. They may have received
                  hypomethylating agents for prior MDS and transfusions, hematopoietic growth
                  factors or vitamins. Temporary prior measures such as apheresis or Hydrea are
                  allowed

               -  For relapsed cohort: Patients with previously treated, relapsed or refractory
                  AML, acute biphenotypic leukemia or high-risk MDS (> 10% bone marrow blasts)

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2

          -  Creatinine < 1.5 mg/dl

          -  Total bilirubin < 1.5 mg/dL, unless increase is due to hemolysis or congenital
             disorder

          -  Transaminases (serum glutamate pyruvate transaminase [SGPT]) < 2.5 x upper limit of
             normal (ULN)

          -  Potassium, magnesium, and calcium (normalized for albumin) levels should be at least
             within institutional normal limits

          -  Ability to take oral medication

          -  Ability to understand and provide signed informed consent

          -  Baseline test of left ventricular ejection fraction >= 50%

          -  Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy
             test within 7 days

          -  WOCBP must use appropriate method(s) of contraception such as oral contraceptive pills
             (OCP), birth control shots, intrauterine device (IUD) etc. WOCBP should use an
             adequate method to avoid pregnancy until 30 days after the last dose of
             investigational drug. Men must agree not to father a child and agree to use a condom
             if his partner is of child bearing potential. Women who are not of childbearing
             potential (ie, who are postmenopausal or surgically sterile) as well as men with known
             azoospermia do not require contraception

        Exclusion Criteria:

          -  Any coexisting medical condition that in the judgment of the treating physician is
             likely to interfere with study procedures or results

          -  Breastfeeding women

          -  Patients with current active malignancies or any remission for < 6 months, except
             patients with carcinoma in situ or with non-melanoma skin cancer who may be in
             remission for less than 6 months or have active disease

          -  Active clinically serious and uncontrolled infection. Patients with recent infections
             must have no temperature of >= 101 degrees Fahrenheit (F) for at least 48 hours (hrs)

          -  Patients with known significant impairment of gastrointestinal (GI) function or GI
             disease that may significantly alter the absorption of quizartinib

          -  Documented active central nervous system leukemia (patients with history of central
             nervous system [CNS] leukemia without active disease are allowed)

          -  Patients with a known confirmed diagnosis of human immunodeficiency virus (HIV)
             infection or active viral hepatitis

          -  Patients who have had any major surgical procedure within 14 days of day 1

          -  Impaired cardiac function including any of the following:

               -  Screening electrocardiography (ECG) with a corrected QT (QTc) > 450 msec. The QTc
                  interval will be calculated by Fridericia's correction factor (QTcF) at screening
                  and prior to the first dose of quizartinib (on day 6 during induction and day 4
                  during consolidation). The QTcF will be derived from the average QTcF in
                  triplicate. If QTcF > 450 msec prior to the first dose, quizartinib will not be
                  given until corrected

               -  Patients with congenital long QT syndrome

               -  Sustained ventricular tachycardia requiring medical intervention

               -  Any history of clinically significant ventricular fibrillation or torsades de
                  pointes

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

               -  Sustained heart rate of < 50/minute on pre-entry ECG

               -  Left bundle branch block

               -  Right bundle branch block + left anterior hemiblock (bifascicular block)

               -  Patients with myocardial infarction or unstable angina within 6 months prior to
                  starting study drug

               -  Congestive heart failure (CHF) New York (NY) Heart Association class III or IV

               -  Atrial fibrillation documented within 2 weeks prior to first dose of study drug

               -  Known family history of congenital long QT syndrome

               -  Patients who are actively taking a strong CYP3A4 inducing medication