In this survey, we provide a selective review of the literature on inflation, innovation, and economic growth. The relationship between economic growth and inflation is a fundamental question in economics. Most studies in this literature explore this relationship in capital-based growth models. This survey reviews a recent branch of this literature on inflation and innovation-driven growth. Specifically, we use a canonical monetary Schumpeterian growth model to demonstrate the effects of inflation on innovation and the macroeconomy via different channels. We find that the cash-in-advance constraints on consumption and R&D investment have drastically different implications on the macroeconomic effects of inflation.