The proteasome inhibitors of this invention include peptide-based compounds with a short linear sequence of amino acids. An oxo or thio group is attached to the N-terminal amino acid. A protein-reactive electrophilic group such as an epoxyketone, an aziridinylketone, or a beta-lactone is attached to the C-terminal amino acid. Upon contact with a proteasome complex in a target cell, the electrophilic group reacts with a functional group in or near a binding pocket or active site of the proteasome, forming a covalent bond and thereby inactivating the proteasome. These and other proteasome inhibitors can be screened for binding affinity and an ability to selectively eliminate senescent cells or cancer cells. Compounds that selectively remove senescent cells can be developed for the treatment of conditions such as osteoarthritis, ophthalmic disease, pulmonary disease, and atherosclerosis.