Genetic constraints are features of inheritance systems that slow or prohibit adaptation. Several population genetic mechanisms of constraint have received sustained attention within the field since they were first articulated in the early 20th century. This attention is now reflected in a rich, and still growing, theoretical literature on the genetic limits to adaptive change. In turn, empirical research on constraints has seen a rapid expansion over the last two decades in response to changing interests of evolutionary biologists, along with new technologies, expanding data sets, and creative analytical approaches that blend mathematical modeling with genomics. Indeed, one of the most notable and exciting features of recent progress in genetic constraints is the close connection between theoretical and empirical research. In this review, we discuss five major population genetic contexts of genetic constraint: genetic dominance, pleiotropy, fitness trade-offs between types of individuals of a population, sign epistasis, and genetic linkage between loci. For each, we outline historical antecedents of the theory, specific contexts where constraints manifest, and their quantitative consequences for adaptation. From each of these theoretical foundations, we discuss recent empirical approaches for identifying and characterizing genetic constraints, each grounded and motivated by this theory, and outline promising areas for future work.