Abstract A new approach for the construction of wavelets and prewavelets on R^d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors from L_2(R^d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multi-resolution.