Cellular protein homeostasis (proteostasis) depends on the controlled degradation of proteins that are damaged or no longer required by the ubiquitin-proteasome system (UPS). The 26S proteasome is the principal executer of substrate-specific proteolysis in eukaryotic cells and regulates a myriad of cellular functions. Proteasome inhibitors were initially developed as chemical tools to study proteasomal function but rapidly became widely used anticancer drugs that are now used at all stages of treatment for the bone marrow cancer multiple myeloma (MM). Here, we review the mechanisms of action of proteasome inhibitors that underlie their preferential toxicity to MM cells, focusing on endoplasmic reticulum stress, depletion of amino acids, and effects on glucose and lipid metabolism. We also discuss mechanisms of resistance to proteasome inhibition such as autophagy and metabolic rewiring and what lessons we may learn from the success and failure of proteasome inhibition in MM for treating other cancers with proteostasis-targeting drugs.