Political competition in democracies is party competition. We develop, from the formal viewpoint, theories of party competition in democracies. The familiar “median voter theorem” of A. Downs is the simplest example of such a theory, but it is inadequate in several ways. We develop a theory in which parties (1) compete over several issues, not just one issue, as in Downs; (2) are uncertain about how citizens will respond to platforms; and (3) represent interest groups in the population. Applications, particularly to the theory of income distribution and taxation, are studied.