Introduction: Difficult Gram-negative infections are increasingly treated with new β-lactamase inhibitor combinations, e.g. ceftazidime/avibactam. Disturbingly, mutations in KPC carbapenemases can confer ceftazidime/avibactam resistance, which is sometimes selected during therapy. We explored whether this risk extended to AmpC and ESBL enzymes. Methods: Mutants were selected by plating AmpC-derepressed strains, ESBL producers and ceftazidime-susceptible controls on agar containing ceftazidime + avibactam (1 or 4 mg/L). MICs were determined by CLSI agar dilution; WGS was by Illumina methodology. Results: Using 2× MIC of ceftazidime + 1 mg/L avibactam, mutants were selected from all strain types at frequencies of 10−7–10−9. Rates diminished to <10−9 with 4 mg/L avibactam or higher MIC multiples, except with AmpC-derepressed Enterobacteriaceae. Characterized mutants (n = 10; MICs 4–64 mg/L) of AmpC-derepressed strains had modifications in ampC, variously giving Arg168Pro/His, Gly176Arg/Asp, Asn366Tyr or small deletions around positions 309–314. Mutants of ESBL producers (n = 19; MICs 0.5–16 mg/L) mostly had changes affecting permeability, efflux or β-lactamase quantity; only one had an altered β-lactamase, with an Asp182Tyr substitution in CTX-M-15, raising the ceftazidime/avibactam MIC, but abrogating other cephalosporin resistance. Mutants of ceftazidime-susceptible strains were not sequenced, but phenotypes suggested altered drug accumulation or, for Enterobacter cloacae only, AmpC derepression. In further experiments, avibactam reduced, but did not abolish, selection of AmpC-derepressed Enterobacteriaceae by ceftazidime. Conclusions: Most mutants of AmpC-derepressed Enterobacteriaceae had structural mutations in ampC; those of ESBL producers mostly had genetic modifications outside β-lactamase genes, commonly affecting uptake, efflux, or β-lactamase quantity. The clinical significance of these observations remains to be determined.