Inclusion Criteria:

          -  Patients must have histologically confirmed, relapsed/refractory ALK+ ALCL (with ALK
             positivity defined by immunohistochemistry and/or fluorescence in situ hybridization
             [FISH]/cytogenetics from any prior biopsy), MCL, or BCL6+ DLBCL (with BCL6 positivity
             defined by immunohistochemistry from any prior biopsy) and meet the following
             criteria:

          -  Patients must have measurable disease that has not been previously irradiated, defined
             as at least one lesion that can be accurately measured in at least one dimension
             (longest diameter to be recorded for non-nodal lesions and short axis for nodal
             lesions) as >= 20 mm (>= 2 cm) with conventional imaging or >= 10 mm with spiral
             computed tomography (CT) scan; if the patient has been previously irradiated, there
             must be evidence of progression since the radiation

               -  Please note, this trial includes mandatory tumor biopsies pre-treatment, during
                  cycle 1 and at the time of disease progression of accessible tumor; having
                  accessible tumor for biopsy is not required for eligibility; we expect that at
                  least 80% of patients will have accessible tumor for these biopsies, however

          -  Prior therapy

               -  Please note, the washout period for prior therapies cannot be shortened

               -  Please note, prior therapies can be from any time in the past

          -  ALK+ ALCL: patients must have disease that has relapsed and or is refractory to prior
             therapy, which must have included a multiagent chemotherapy regimen including an
             anthracycline, if not contraindicated, and prior brentuximab; prior crizotinib or
             other ALK inhibitor therapy, while recommended, is not mandatory; patients must have
             relapsed following or be ineligible for, or refuse, autologous stem cell transplant

          -  MCL: patients must have disease that has relapsed and or is refractory to prior
             therapy, which must have included a multiagent chemotherapy regimen and prior
             ibrutinib or other BTK inhibitor therapy; patients must have relapsed following or be
             ineligible for, or refuse, autologous stem cell transplant

          -  BCL6+ DLBCL: patients must have disease that has relapsed and or is refractory to
             prior therapy, which must have included an anthracycline, if not contraindicated;
             patients must have relapsed following or be ineligible for, or refuse, autologous stem
             cell transplant

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Life expectancy of greater than 3 months

          -  Absolute neutrophil count >= 1,000/mcL

          -  Platelets >= 75,000/mcL, unless due to marrow involvement by lymphoma in which case a
             platelet count of >= 30,000/mcL will be used

          -  Total bilirubin =< 1.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome
             or hemolysis, in which case =< 3.0 x ULN is allowed

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 3.0 x institutional upper limit of normal

          -  Creatinine =< 1.5 x ULN or a creatinine clearance >= 50 mL/min/1.73 m^2 for patients
             with creatinine levels above institutional normal

          -  Potassium above the institutional lower limit of normal (supplementation to meet this
             is allowed)

          -  Magnesium above the institutional lower limit of normal (supplementation to meet this
             is allowed)

          -  Human immunodeficiency virus (HIV)+ patients are eligible for the trial provided they
             meet the other study criteria in addition to the following:

               -  CD4+ T-cells >= 250/mm^3

               -  HIV sensitive to antiretroviral therapy

               -  Zidovudine not allowed

               -  Long term survival anticipated on the basis of HIV alone were it not for the
                  lymphoma

               -  No concurrent acquired immunodeficiency syndrome (AIDS)-defining illness other
                  than the lymphoma

          -  The effects of AT13387 (onalespib) on the developing human fetus are unknown; for this
             reason, women of child-bearing potential and men 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 4 months after the completion
             of AT13387 (onalespib) administration; should a woman become pregnant or suspect she
             is pregnant while she or her partner is participating in this study, she should inform
             her treating physician immediately; men treated or enrolled on this protocol must also
             agree to use adequate contraception prior to the study, for the duration of study
             participation, and 4 months after completion of AT13387 (onalespib) administration

          -  Although the pharmacokinetic (PK) data in humans is still unknown, the potential for
             drug-interaction cannot be ruled out; pre-clinical studies suggest that AT13387
             (onalespib) is a substrate of P-glycoprotein (P-gp), a moderate inhibitor of BCRP and
             P-gp, and a strong inhibitor of MATE 1/2-K; patients must be willing to not take St.
             John wort or grapefruit juice while participating in this trial and should avoid drugs
             that are strong inducers of P-gp, and to switch to alternative drugs when available

          -  Hepatitis B positive patients are eligible; prophylactic hepatitis B virus (HBV)
             therapy is recommended but only if there is no circulating virus detectible

          -  Transformed lymphoma patients are eligible

          -  Ability to understand and the willingness to sign a written informed consent document

        Exclusion Criteria:

          -  Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for
             nitrosoureas or mitomycin C) prior to entering the study or those who have not
             recovered from adverse events due to agents administered more than 4 weeks earlier;
             steroids for symptom palliation are allowed, but must be either discontinued or on
             stable doses at the time of initiation of protocol therapy

          -  Patients who are receiving any other investigational agents; all investigational
             agents other than ibrutinib must have been discontinued at least 4 weeks prior to
             beginning treatment; prior ibrutinib therapy must have been discontinued at least 2
             weeks prior to beginning therapy

          -  Patients with known leptomeningeal or brain metastases should be excluded from this
             clinical trial because of their poor prognosis and because they often develop
             progressive neurologic dysfunction that would confound the evaluation of neurologic
             and other adverse events; imaging or spinal fluid analysis to exclude central nervous
             system (CNS) involvement is not required, unless there is clinical suspicion by the
             treating investigator

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to AT13387 (onalespib)

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements

               -  There will be no exclusion of patients with known visual impairment or symptoms,
                  including by not limited to peripheral flashes (photopsia), blurred or double
                  vision, floaters, color distortion and dimness, difficulties with light/dark
                  accommodation, tunnel vision or other field defects, halos, apparent movement of
                  stationary objects, and complex disturbances; patients will have a baseline
                  ophthalmologic exam to serve as a point of comparison and further exams as needed
                  should visual symptoms develop; no pretreatment eye exam findings or ocular
                  symptoms have been associated with an increased risk of ocular toxicity seen with
                  AT13387

          -  Pregnant women are excluded from this study because AT13387 (onalespib) has the
             potential for teratogenic or abortifacient effects; because there is an unknown but
             potential risk for adverse events in nursing infants secondary to treatment of the
             mother with AT13387 (onalespib), breastfeeding should be discontinued if the mother is
             treated with AT13387 (onalespib)

          -  Patients, who have had a major surgery or significant traumatic injury within 4 weeks
             of start of study drug, patients who have not recovered from the side effects of any
             major surgery (defined as requiring general anesthesia) or patients that may require
             major surgery during the course of the study

          -  Prior history of another malignancy (except for non-melanoma skin cancer or in situ
             cervical or breast cancer) unless disease free for at least three years; patients with
             prostate cancer are allowed if prostate specific antigen (PSA) is less than 1

          -  Patients should not receive immunization with attenuated live vaccine within one week
             of study entry or during study period

          -  History of noncompliance to medical regimens

          -  Consistent corrected QT (QTc) > 450 msec for men and > 470 msec for women by
             Fridericia formula, on 3 separate electrocardiograms (ECGs)

          -  Left ventricular ejection fraction (LVEF) < 50%, regardless of whether there are
             symptoms of heart failure