Summary: We introduce a new model for time-varying conditional variances as the most general quadratic version possible within the ARCH class. Hence, it encompasses all the existing restricted quadratic variance functions. Its properties are very similar to those of GARCH models, but avoids some of their criticisms. In univariate applications to daily U.S. and monthly U.K. stock market returns, QARCH adequately represents volatility and risk premia. QARCH is easy to incorporate in multivariate models to capture dynamic asymmetries that GARCH rules out. Such asymmetries are found in an empirical application of a conditional factor model to 26 U.K. sectorial stock returns.