We review the one-dimensional setting of wavelet theory and generalize it to nonseparable multivariate wavelets. This process presents significant technical difficulties. Some techniques of the one-dimensional setting carry over in a more or less straightforward way; some do not generalize at all.;The main results include the following: an algorithm for computing the moments for multivariate multiwavelets; a necessary and sufficient condition for the approximation order; the lifting scheme for multivariate wavelets; and a generalization of the method of Lai [12] for the biorthogonal completion of a polyphase matrix under suitable conditions.;One-dimensional techniques which cannot be generalized include the factorization of the polyphase matrix, and a general solution to the completion problem.