Wavelets have been playing a prominent role in approximation theory, signal processing, and many other different contexts for several decades. A lot of effort has been spent on designing wavelet bases, in particular in the multivariate setting. Nevertheless, general procedures for their construction remain challenging tasks. This paper aims to illustrate and summarize the research carried out in the last few years by the authors and collaborators in the direction of exploring new strategies for realizing novel wavelet filterbank systems but also of unifying different existing approaches, both in the univariate and in the bivariate cases.