Anti-HIV broadly neutralizing antibodies (bnAbs) have revealed vaccine targets on the virus's envelope (Env) protein and are themselves promising immunotherapies. The efficacy of bnAb-based therapies and vaccines depends in part on how readily the virus can escape neutralization. Although structural studies can define contacts between bnAbs and Env, only functional studies can define mutations that confer escape. Here, we mapped how all possible single amino acid mutations in Env affect neutralization of HIV by nine bnAbs targeting five epitopes. For most bnAbs, mutations at only a small fraction of structurally defined contact sites mediated escape, and most escape occurred at sites near, but not in direct contact with, the antibody. The Env mutations selected by two pooled bnAbs were similar to those expected from the combination of the bnAbs's independent action. Overall, our mutation-level antigenic atlas provides a comprehensive dataset for understanding viral immune escape and refining therapies and vaccines.