A PCR primer sequence is called degenerate if some of its positions have several possible bases. The degeneracy of the primer is the number of unique sequence combinations it contains. We study the problem of designing a pair of primers with prescribed degeneracy that match a maximum number of given input sequences. Such problems occur when studying a family of genes that is known only in part, or is known in a related species. We prove that various simplified versions of the problem are hard, show the polynomiality of some restricted cases, and develop approximation algorithms for one variant. Based on these algorithms, we implemented a program called HYDEN for designing highly degenerate primers for a set of genomic sequences. We report on the success of the program in several applications, one of which is an experimental scheme for identifying all human olfactory receptor (OR) genes. In that project, HYDEN was used to design primers with degeneracies up to 10(10) that amplified with high specificity many novel genes of that family, tripling the number of OR genes known at the time.