Inclusion Criteria:

          -  Patients must have histologically confirmed diagnosis of Philadelphia-negative ALL by
             bone marrow biopsy or aspirate

          -  Patients must have >= 5% blasts in the bone marrow

          -  Patients must have refractory disease, disease relapse or progression after at least
             two prior systemic chemotherapy or immunotherapy regimens

               -  Note: Exceptions may be made if a patient is deemed unfit for first-line salvage
                  therapy by the treating physician; such cases should be clearly documented

          -  Patients with a history of CNS (central nervous system) leukemia are eligible if they
             are not symptomatic from current CNS involvement; if there is CNS involvement that is
             known prior to enrollment or identified subsequently, it will be treated accordingly
             with intrathecal chemotherapy per the treating physician

          -  Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status
             of =< 2

          -  Patients must have adequate organ function within 14 days prior to registration, as
             defined below:

          -  Total bilirubin =< 2 x institutional upper limit of normal (ULN)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/aspartate aminotransferase (ALT) (serum glutamic pyruvic transaminase [SPGT])
             =< 3 x ULN

          -  Creatinine =< 2 X ULN

          -  Females of child-bearing potential (FOCBP) and males must agree to use adequate
             contraception (hormonal or barrier method of birth control; abstinence) prior to study
             entry, for the duration of study participation, and for 6 months following completion
             of therapy; should a female patient become pregnant or suspect she is pregnant while
             participating in this study, she should inform her treating physician immediately

               -  NOTE: A FOCBP is any woman (regardless of sexual orientation, having undergone a
                  tubal ligation, or remaining celibate by choice) who meets the following
                  criteria:

                    -  Has not undergone a hysterectomy or bilateral oophorectomy

                    -  Has had menses at any time in the preceding 12 consecutive months (and
                       therefore has not been naturally postmenopausal for > 12 months)

          -  FOCBP must have a negative pregnancy test within 14 days prior to registration on
             study

          -  Patients must have the ability to understand and the willingness to sign a written
             informed consent prior to registration on study

        Exclusion Criteria:

          -  Patients who have had chemotherapy, immunotherapy, or radiotherapy within 2 weeks
             prior to entering the study or those who have not recovered from adverse events (=<
             grade 1 or patient's baseline) due to agents administered more than 2 weeks earlier
             are not eligible

          -  Patients may not be receiving any other investigational agents within 7 days of
             registration

          -  Patients may not be receiving any medications that are known to prolong QT interval
             unless reviewed and approved by the principal investigator (PI)

          -  Patients who have a history of allergic reactions attributed to compounds of similar
             chemical or biologic composition to volasertib or VSLI are not eligible

          -  Subject may not have had hematopoietic stem cell transplant (HSCT) meeting any of the
             following:

               -  Is within 2 months of transplant from cycle 1 day 1 (C1D1)

               -  Has clinically significant graft-versus-host disease requiring treatment

               -  Has >= grade 2 persistent non-hematological toxicity related to the transplant

               -  Donor lymphocyte infusion (DLI) is not permitted < 30 days prior to study
                  registration

          -  Patients with >= grade 2 sensory or motor neuropathy are not eligible

          -  Fridericia's corrected QT (QTcF) prolongation > 470 ms or QT prolongation deemed
             clinically relevant by the investigator (e.g., congenital long QT syndrome); the QTcF
             will be calculated as the mean of the 3 electrocardiograms (ECGs) taken at screening

               -  NOTE: The formula used to calculate QTcF can be physician's choice, but it must
                  be used consistently throughout the study

          -  Patients who have an uncontrolled intercurrent illness including, but not limited to
             any of the following, are not eligible:

               -  Ongoing or active infection requiring systemic treatment

               -  Symptomatic congestive heart failure (>= class 3)

               -  Unstable angina pectoris

               -  Cardiac arrhythmia

               -  Psychiatric illness/social situations that would, in the investigator's opinion,
                  limit compliance with study requirements

               -  Known human immunodeficiency virus (HIV) infection

               -  Known John Cunningham virus (JC) virus infection and/or progressive multifocal
                  leukoencephalopathy (PML)

               -  Known clinically active hepatitis A, B, or C infections

                    -  NOTE: Patients with chronic hepatitis C virus (HCV) or hepatitis B virus
                       (HBV) infection may enroll if other laboratory criteria are met; those with
                       HBV surface antigen positivity may enroll only if maintained on appropriate
                       suppressive antiviral therapy for the duration of enrollment in the trial

               -  Second malignancy that requires active treatment

               -  Any other illness or condition that the treating investigator feels would
                  interfere with study compliance or would compromise the patient's safety or study
                  endpoints

          -  Female patients who are pregnant or nursing are not eligible