CRISPR–Cas is a widespread adaptive immune system in bacteria and archaea that protects against viral infection by targeting specific invading nucleic acid sequences. Whereas some CRISPR–Cas systems sense and cleave viral DNA, type III and type VI CRISPR–Cas systems sense RNA that results from viral transcription and perhaps invasion by RNA viruses. The sequence-specific detection of viral RNA evokes a cell-wide response that typically involves global damage to halt the infection. How can one make sense of an immune strategy that encompasses broad, collateral effects rather than specific, targeted destruction? In this Review, we summarize the current understanding of RNA-targeting CRISPR–Cas systems. We detail the composition and properties of type III and type VI systems, outline the cellular defence processes that are instigated upon viral RNA sensing and describe the biological rationale behind the broad RNA-activated immune responses as an effective strategy to combat viral infection.