RNA-guided programmable nucleases from CRISPR systems generate precise breaks in DNA or RNA at specified positions. In cells, this activity can lead to changes in DNA sequence or RNA transcript abundance. Base editing is a newer genome-editing approach that uses components from CRISPR systems together with other enzymes to directly install point mutations into cellular DNA or RNA without making double-stranded DNA breaks. DNA base editors comprise a catalytically disabled nuclease fused to a nucleobase deaminase enzyme and, in some cases, a DNA glycosylase inhibitor. RNA base editors achieve analogous changes using components that target RNA. Base editors directly convert one base or base pair into another, enabling the efficient installation of point mutations in non-dividing cells without generating excess undesired editing by-products. In this Review, we summarize base-editing strategies to generate specific and precise point mutations in genomic DNA and RNA, highlight recent developments that expand the scope, specificity, precision and in vivo delivery of base editors and discuss limitations and future directions of base editing for research and therapeutic applications.