Identifying viral mutations that confer escape from antibodies is crucial for understanding the interplay between immunity and viral evolution. We describe a high-throughput approach to quantify the selection that monoclonal antibodies exert on all single amino-acid mutations to a viral protein. This approach, mutational antigenic profiling, involves creating all replication-competent protein variants of a virus, selecting with antibody, and using deep sequencing to identify enriched mutations. We use mutational antigenic profiling to comprehensively identify mutations that enable influenza virus to escape four monoclonal antibodies targeting hemagglutinin, and validate key findings with neutralization assays. We find remarkable mutation-level idiosyncrasy in antibody escape: for instance, at a single residue targeted by two antibodies, some mutations escape both antibodies while other mutations escape only one or the other. Because mutational antigenic profiling rapidly maps all mutations selected by an antibody, it is useful for elucidating immune specificities and interpreting the antigenic consequences of viral genetic variation.