We define generalized Hamming weights for almost affine codes. We show how various aspects and applications of generalized Hamming weights for linear codes, such as Wei duality, generalized Kung's bound, profiles, connection to wire-tap channels of type II, apply to the larger class of almost affine codes in general. In addition we discuss duality of almost affine codes,and of the smaller class of multilinear codes. We also give results about weight distributions of infinite series of almost affine codes, each series obtained from a fixed code by extending the code alphabet.