Inclusion Criteria: - Ability to understand the risks of the study and provide signed informed consent - Age 18 years or older - Relapsed and or Refractory MM - Disease progression following at least two systemic treatments for MM - ECOG performance status 0, 1 or 2 Exclusion Criteria: - Pregnant or lactating females. Patients of childbearing potential must use appropriate birth control during the entire duration of the study - Inadequate liver function - Renal impairment - Neutrophil count <1.0 x 109 /litre in the absence of growth factors - Platelet count <50 x 109 /litre in patients in whom <50% of bone marrow nucleated cells are plasma cells and <30 x 109 /litre in patients in whom ≥50 % of bone marrow nucleated cells are plasma cells - Hemoglobin <8g/dl in the absence of transfusion - Treated corrected calcium >ULN - Serum creatine phosphokinase >ULN - All previous cytotoxic therapies for MM must have been completed at least four weeks prior to treatment with AT7519M (two weeks for all non-cytotoxic therapy) - Prior peripheral stem cell transplant within 12 weeks - Evidence of mucosal or internal bleeding and/or platelet transfusion refractory (unable to maintain a platelet count >50 x 109 /litre) - Ongoing infection requiring treatment - Previous radiotherapy within 2 weeks of the start of the study - Having previously received treatment with a cyclin-dependent kinase or GSK3beta inhibitor - Incomplete recovery from previous radiotherapy other than residual cutaneous effects or stable < Grade 2 gastrointestinal toxicity - Prior radiotherapy to the head and neck region - Previous malignancy, except for non-basal-cell carcinoma of skin, carcinoma-in-situ of the uterine cervix, ductal-carcinoma in situ - Any severe or uncontrolled systemic conditions (e.g., systemic infection) or current unstable or uncompensated respiratory or cardiac conditions which makes it undesirable for the patient to participate in the study or which could jeopardize compliance with the protocol - Incomplete recovery from surgery other than stable < Grade 2 toxicity - Peripheral neuropathy > Grade 2 - Abnormal left ventricular ejection fraction (< 45 %) on echocardiogram - History of an ischemic cardiac event, including myocardial infarction within 3 months of study entry - Congestive cardiac failure of ≥ grade 3 severity according to NYHA functional classification - Unstable cardiac disease - History or presence of bradycardia (≤60bpm), left bundle branch block, heart block, cardiac pacemaker or significant atrial tachyarrythmias - Concurrent treatment with any medication known strongly to inhibit or induce CYP3A4, CYP1A2 and CYP2C19 which cannot be discontinued at least two week prior to treatment with AT7519M (other than corticosteroids) - Concurrent treatment with any medication that prolongs QT interval and may induce Torsades de Pointes and which cannot be discontinued at least two weeks prior to treatment with AT7519M - Family or personal history of long QTc syndrome or ventricular arrhythmias including ventricular bigeminy - Previous history of drug-induced QTc prolongation - Screening 12-lead ECG with measurable QTc interval according to Fridericia's Correction of >450 msecs - Screening 12-lead ECG with ST depression >1 mm in 2 or more leads or T wave inversion in 2 or more contiguous leads - Prior history of infection with human immunodeficiency virus (HIV), known active hepatitis B or C viruses - Diffuse infiltrative pulmonary or pericardial disease - Known hypersensitivity to bortezomib, boron or any of the excipients of VelcadeTM - Epilepsy or other convulsive disorder requiring active management - Serious psychiatric illness, active alcoholism or drug addiction that may hinder or confuse follow up evaluation