Identification based on higher moments has drawn increasing theoretical attention and been widely adopted in empirical practice in macroeconometrics in the last two decades. This article reviews two parallel strands of the literature: identification strategies based on heteroskedasticity and strategies based on non-Gaussianity more generally. I outline the seminal identification results and discuss recent extensions, parametric and non-parametric implementations, and prominent empirical applications. I additionally describe key issues for the adoption of such strategies, including weak identification and interpretability of statistically identified structural shocks. I further outline key areas of ongoing research.