There are powerful imperatives on governments to support innovation, which have only grown more urgent amid slowing growth in the developed world and a rapidly changing climate. In this review, I describe the important role that economics and finance play in rigorously evaluating innovation policy. I organize government intervention in innovation into arenas, agendas, and instruments. The arenas are firms, financial intermediaries, universities, and government laboratories. The key agendas are entrepreneurship, defense, climate, health, and education. The instruments fall into three categories: supply-push, demand-pull, and legal. I provide theoretical rationales for government intervention in innovation and discuss how they intersect with government agendas and empirical evidence in practice. One takeaway is that the government has a key role to play in the type of failure-tolerant, open-ended research that yields breakthrough inventions. In contrast, there is less evidence that subsidizing financial intermediaries is useful.