Base editors use DNA-modifying enzymes targeted with a catalytically impaired CRISPR protein to precisely install point mutations. Here, we develop phage-assisted continuous evolution of base editors (BE-PACE) to improve their editing efficiency and target sequence compatibility. We used BE-PACE to evolve cytosine base editors (CBEs) that overcome target sequence context constraints of canonical CBEs. One evolved CBE, evoAPOBEC1-BE4max, is up to 26-fold more efficient at editing cytosine in the GC context, a disfavored context for wild-type APOBEC1 deaminase, while maintaining efficient editing in all other sequence contexts tested. Another evolved deaminase, evoFERNY, is 29% smaller than APOBEC1 and edits efficiently in all tested sequence contexts. We also evolved a CBE based on CDA1 deaminase with much higher editing efficiency at difficult target sites. Finally, we used data from evolved CBEs to illuminate the relationship between deaminase activity, base editing efficiency, editing window width and byproduct formation. These findings establish a system for rapid evolution of base editors and inform their use and improvement.