Cas12a is a bacterial RNA-guided nuclease used widely for genome editing and, more recently, as a molecular diagnostic. In bacteria, Cas12a enzymes can be inhibited by bacteriophage-derived proteins, anti-CRISPRs (Acrs), to thwart clustered regularly interspaced short palindromic repeat (CRISPR) adaptive immune systems. How these inhibitors disable Cas12a by preventing programmed DNA cleavage is unknown. We show that three such inhibitors (AcrVA1, AcrVA4 and AcrVA5) block Cas12a activity via functionally distinct mechanisms, including a previously unobserved enzymatic strategy. AcrVA4 and AcrVA5 inhibit recognition of double-stranded DNA (dsDNA), with AcrVA4 driving dimerization of Cas12a. In contrast, AcrVA1 is a multiple-turnover inhibitor that triggers cleavage of the target-recognition sequence of the Cas12a-bound guide RNA to irreversibly inactivate the Cas12a complex. These distinct mechanisms equip bacteriophages with tools to evade CRISPR-Cas12a and support biotechnological applications for which multiple-turnover enzymatic inhibition of Cas12a is desirable.