A class of nonlinear ARCH models is suggested. The proposed class encompasses several functional forms for ARCH which have been put forth in the literature. A Lagrange multiplier test is developed to test Engle's ARCH specification against the wider class of models. This test provides an easily computed diagnostic check of the adequacy of an ARCH model after it has been estimated. The theory is applied to a number of weekly exchange rate series and we find strong evidence of nonlinear ARCH.