Influenza remains a major human pathogen despite seasonal vaccination. At long last, there is energy and resources to develop influenza vaccines that provide more predictable and durable protection. Vaccines based on inducing antibodies to the conserved stem of the viral hemagglutinin (HA) have emerged as leading candidates for broadening population immunity and ultimately limiting antigenic drift. Here, we discuss the knowns and unknowns of HA-specific B-cell and antibody responses. In particular, we focus on how immunodominance sculpts antibody responses and drives antigenic drift. We propose a number of strategies to overcome immunodominance and improve the breadth and efficacy of antibody responses.