Identifying viral mutations that confer escape from antibodies is crucial for understanding the interplay between immunity and viral evolution. Here we quantify how every amino-acid mutation to influenza hemagglutinin affects neutralization by monoclonal antibodies targeting several antigenic regions. Our approach involves creating all replication-competent protein variants of the virus, selecting these variants with antibody, and using deep sequencing to identify enriched mutations. These high-throughput measurements are predictive of the effects of individual mutations in traditional neutralization assays. At many residues, only some of the possible mutations escape from an antibody. For instance, at a single residue targeted by two different antibodies, we identify some mutations that escape both antibodies and other mutations that escape only one or the other. Therefore, our approach maps how viruses can escape antibodies with mutation-level sensitivity, and shows that only some mutations at antigenic residues actually alter antigenicity.