The purpose of this open-label, non-randomized trial is to assess the safety and effectiveness of PXD101, both alone and in combination with dexamethasone, in patients with advanced multiple myeloma. PXD101 is a new, potent histone deacetylase (HDAC) inhibitor. Various members of this class of drugs have shown activity in preclinical studies and in initial clinical trials of multiple myeloma and lymphoma. Furthermore, HDAC inhibitors, including PXD101, have been shown to sensitize myeloma cells to the killing effect of other chemotherapeutic agents, including dexamethasone, a well-established agent in relapsing myeloma.