A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition of L 2(R)s,s s1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis of L 2(R)s 1s3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for "truncated" decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. 1992 Springer.