Stress granules and P bodies are conserved cytoplasmic aggregates of nontranslating messenger ribonucleoprotein complexes (mRNPs) implicated in the regulation of mRNA translation and decay and are related to RNP granules in embryos, neurons, and pathological inclusions in some degenerative diseases. Using baker’s yeast, 125 genes were identified in a genetic screen that affected the dynamics of P bodies and/or stress granules. Analyses of such mutants, including CDC48 alleles, provide evidence that stress granules can be targeted to the vacuole by autophagy, in a process termed granulophagy. Moreover, stress granule clearance in mammalian cells is reduced by inhibition of autophagy or by depletion or pathogenic mutations in valosin-containing protein (VCP), the human ortholog of CDC48. Because mutations in VCP predispose humans to amyotrophic lateral sclerosis, frontotemporal lobar degeneration, inclusion body myopathy, and multisystem proteinopathy, this work suggests that autophagic clearance of stress granule related and pathogenic RNP granules that arise in degenerative diseases may be important in reducing their pathology. Display omitted •Network of 125 genes identified affecting stress-granule and P body dynamics in yeast•Stress granules are cleared by autophagy in yeast and mammalian cells•Cdc48/VCP facilitates stress-granule clearance•Autophagy of related mRNP aggregates may help avoid degenerative pathology Proteins linked to modulating stress-granule levels include key autophagy factors, and regulation of these RNP aggregates by autophagic clearance, involving VCP/Cdc48, is conserved in yeast and mammalian cells; this has implications for addressing these kinds of aggregates in neuronal pathologies.