CRISPR-Cas12a (Cpf1) is an RNA-guided DNA-cutting nuclease that has been repurposed for genome editing. Upon target DNA binding, Cas12a cleaves both the target DNA in cis and non-target single-stranded DNAs (ssDNAs) in trans. To elucidate the molecular basis for both DNase cleavage modes, we performed structural and biochemical studies on Francisella novicida Cas12a. We show that guide RNA-target strand DNA hybridization conformationally activates Cas12a, triggering its trans-acting, non-specific, single-stranded DNase activity. In turn, cis cleavage of double-stranded DNA targets is a result of protospacer adjacent motif (PAM)-dependent DNA duplex unwinding, electrostatic stabilization of the displaced non-target DNA strand, and ordered sequential cleavage of the non-target and target DNA strands. Cas12a releases the PAM-distal DNA cleavage product and remains bound to the PAM-proximal DNA cleavage product in a catalytically competent, trans-active state. Together, these results provide a revised model for the molecular mechanisms of both the cis- and the trans-acting DNase activities of Cas12a enzymes, enabling their further exploitation as genome editing tools. Display omitted •Target ssDNA binding allosterically induces unblocking of the RuvC active site•PAM binding facilitates strand separation in dsDNA targets•Non-target DNA strand cleavage precedes target DNA strand cleavage•Cas12a releases the PAM-distal end of cleaved dsDNA targets Swarts et al. elucidated two structures of Cas12a in complex with a guide RNA and with either single-stranded or double-stranded DNA targets. Together with biochemical assays, these structures reveal the mechanisms of Cas12a-mediated cis and trans cleavage of DNA substrates.