We first characterize the Riesz wavelets which are associated with multiresolution analyses (MRAs) and the Riesz wavelets whose duals are also Riesz wavelets. The characterizations show that if a Riesz wavelet is associated with an MRA, then it has a dual Riesz wavelet. We then improve Wang's characterization for a pair of biorthogonal wavelets to be associated with biorthogonal MRAs by showing that one of the two conditions in his characterization is redundant. (C) 2001 Academic Press.