How do voters allocate costly attention to alternative political issues? And how does selective ignorance of voters interact with policy design by politicians? We address these questions by developing a model of electoral competition with rationally inattentive voters. Rational inattention amplifies the effects of preference intensity, because voters pay more attention where stakes are higher. The model has many potential applications, and those that we discuss in more detail imply that extremist voters are more attentive and inuential, public goods are under-provided, divisive issues receive more attention, and less transparent candidates choose more extreme policies. Endogenous attention can also lead to multiple equilibria, explaining how poor voters in developing countries can be politically empowered by welfare programs.