Meyer, Sellan and Taqqu (1999, JFAA, “Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion”) established an interesting wavelet-based decomposition of fractional Brownian motion. With a suitable choice of a multiresolution wavelet basis, the wavelet coefficients in the decomposition were independent Gaussian variables and the low frequency terms in the decomposition involved, for example, partial sums of stationary FARIMA series as a natural approximation to fractional Brownian motion. In this talk, I will describe extensions of this result to the Rosenblatt process (an important example of a non-Gaussian self-similar process with stationary increments), and quite general Gaussian stationary processes (thus not self-similar). I will also discuss several applications (such as synthesis and estimation) and other issues (such as the Riesz property of the wavelet-like bases involved). Parts of this talk are based on joint works with P. Abry (ENS, Lyon) and G. Didier (Tulane University).