GARCH models for time-varying volatility

What is GARCH and why is it used for financial returns?

Generalised autoregressive conditional heteroskedasticity: a model where variance depends on past shocks and past variance. The basic form is $\sigma_t^2=\omega+\alpha\epsilon_{t-1}^2+\beta\sigma_{t-1}^2$. It captures volatility clustering and fat-tailed unconditional returns and gives volatility forecasts that mean-revert to a long-run level; variants such as EGARCH and GJR add asymmetry so falls raise volatility more than rises.