We extend the class of GARCH models to comprise asymmetric and nonlinear effects on volatility. In particular, we do not only explain future volatility of a time series on its own past, but allow for external influences and spillovers between capital markets. For this generalized class of models, the asymptotic behavior of the Quasi-Maximum-Likelihood estimator of model parameters is derived. The models are applied to time series of fx-rates. It is found that in particular the simple asymmetric models lead to improved performance.