This phase I/Ib trial studies the side effects and best dose of panobinostat and carfilzomib in treating participants with multiple myeloma that has come back or that isn't responding to treatment. Carfilzomib keeps cancer cells from repairing themselves. If the cancer cells cannot repair themselves, they may die. Panobinostat may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Giving panobinostat and carfilzomib may work better in treating participants with multiple myeloma.

PRIMARY OBJECTIVES:

I. To determine the maximum recommended dose (MRD) of the combination of carfilzomib and panobinostat in patients with relapsed/refractory multiple myeloma (RRMM). (Phase I) II. To determine the overall response rate (stable disease [SD], minimal response [MR], partial response [PR], very good partial response [VGPR], near complete response/complete response [nCR/CR] of the combination. (Phase Ib)

SECONDARY OBJECTIVES:

I. To determine the overall response rate (ORR), stable disease (SD), minimal response (MR), partial response (PR), very good partial response (VGPR), near complete response/complete response (nCR/CR). (Phase I) II. To determine the time to progression (TTP). (Phase I and Ib) III. To determine the progression free survival (PFS). (Phase I and Ib) IV. To determine the time to best response. (Phase I and Ib) V. To determine time to next therapy. (Phase I and Ib) VI. To determine duration of response. (Phase I and Ib) VII. To determine the overall response rate (SD, MR, PR, VGPR, nCR/CR) of the combination. (Phase Ib) VIII. To assess the safety of the combination. (Phase Ib)

OUTLINE: This is a phase I and Ib dose escalation study.

Participants receive carfilzomib intravenously (IV) over 30 minutes on days 1, 2, 8, 9, 15, and 16 and panobinostat orally (PO) once daily (QD) on days 1, 3, 5, 8, 10, and 12 of each course. Courses repeat every 28 days for up to 8 courses in the absence of disease progression or unacceptable toxicity. After 8 courses, participants may continue carfilzomib IV on days 1, 2, 15, and 16, and panobinostat PO on days 1, 3, 5, 8, 10, and 12 of each course. If the disease becomes worse, participants can receive carfilzomib on the original dosing schedule (days 1, 2, 8, 9, 15, and 16 of each course).

After completion of study treatment, participants are followed up at 30 days.

Inclusion Criteria:

1. Relapsed/refractory MM with failure to at least two lines of MM treatment which must include at least one IMiD (thalidomide, lenalidomide) and proteosome inhibitor (bortezomib) and measurable levels of myeloma paraprotein in serum ( >/= 0.5 g/dl), urine ( >/= 0.2 g excreted in a 24-hour collection sample), or abnormal free light chain (FLC) ratio. Oligo or non secretory myeloma patients may be included, if there is measurable plasmacytosis in the bone marrow biopsy or measurable extramedullary disease.
2. Male or female patients aged >/= 18 years old
3. Ability to provide written informed consent obtained prior to participation in the study and any related procedures being performed
4. Patients must meet the following laboratory criteria within 28 days of starting therapy: * Absolute neutrophil count (ANC) >/= 1.0 x 10^9/L * Hemoglobin >/= 8 g/dl ( transfusion are permitted) * Platelet count > 70,000 cells/mm^3 for patients with < 50% of bone marrow plasma cells or platelet count > 25,000 cells/mm^3 for patients in whom > 50% of the bone marrow nucleated cells were plasma cells * aspartate aminotransferase (AST) and Alanine aminotranferease (ALT) </= 2.5 x ULN * Serum bilirubin </= 2 x ULN
5. ECOG Performance Status of </= 2
6. Creatinine clearance (CrCl) >/= 30 mL/minute within 28 days prior to registration, either measured or calculated using a standard formula (eg, Cockcroft and Gault)
7. Multiple Gated Acquisition (MUGA) or echocardiogram (ECHO) must demonstrate LVEF >/= 45%.
8. Female patients who: Are postmenopausal for at least 1 year before the screening visit, OR Are surgically sterile, OR If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent through 30 days after the last dose of study drug, or agree to completely abstain from heterosexual intercourse. Male patients, even if surgically sterilized (ie, status postvasectomy), who: Agree to practice effective barrier contraception during the entire study treatment period and through 30 days after the last dose of study drug, OR Agree to completely abstain from heterosexual intercourse.

Exclusion Criteria:

1. Valproic acid for the treatment of cancer
2. Patients who will need valproic acid for any medical condition during the study or within 5 days prior to first panobinostat treatment
3. Impaired cardiac function or clinically significant cardiac diseases, including any one of the following: * History or presence of sustained ventricular tachyarrhythmia. (Patients with a history of atrial arrhythmia are eligible) * Any history of ventricular fibrillation or torsade de pointes * Bradycardia defined as heart rate (HR)< 50 bpm. Patients with pacemakers are eligible if HR >/= 50 bpm. * Screening electrocardiogram (ECG) with a corrected QT interval (QTc) or QTcF > 450 msec * Right bundle branch block + left anterior hemiblock (bifascicular block) * Patients with myocardial infarction or unstable angina </= 6 months prior to starting study drug * Other clinically significant heart disease (e.g., Congestive Heart Failure (CHF) New York Heart Association (NYHA) class III or IV , uncontrolled hypertension, history of labile hypertension, or history of poor compliance with an antihypertensive regimen)
4. Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of panobinostat
5. Patients with diarrhea > Common Terminology Criteria for Adverse Events (CTCAE) grade 2
6. Other concurrent severe and/or uncontrolled medical conditions (e.g., uncontrolled diabetes or active or uncontrolled infection) including abnormal laboratory values, that could cause unacceptable safety risks or compromise compliance with the protocol
7. Patients using medications that have a relative risk of prolonging the QT interval or inducing torsade de pointes if treatment cannot be discontinued or switched to a different medication prior to starting study drug
8. Patients who have received chemotherapy within </= 2 weeks by time of cycle 1 day 1 of therapy on trial ; or radiation therapy to > 30% of marrow-bearing bone within 2 weeks prior to starting study treatment; or who have not yet recovered from side effects of such therapies.
9. Female patients who are lactating or have a positive serum or urine pregnancy test during the Screening period.
10. Patients with any significant history of non-compliance to medical regimens or unwilling or unable to comply with the instructions given to him/her by the study staff.
11. Known history of allergy to Captisol® (a cyclodextrin derivative used to solubilize carfilzomib)