Blind source separation (BSS) is an active area of research due to its applicability to a variety of problems, especially when there is a little known about the observed data. Applications where BSS has been successfully utilized include the analysis of medical imaging data, such as functional magnetic resonance imaging (fMRI) data, detection of specific targets in video sequences or multi-spectral remote sensing data, among many others. Independent component analysis (ICA) is a widely used BSS method that can uniquely achieve source recovery, subject to only scaling and permutation ambiguities, through the assumption of statistical independence on the part of the latent sources. Independent vector analysis (IVA) extends the applicability of ICA by jointly decomposing multiple datasets through the exploitation of the dependencies across datasets. Though both ICA and IVA algorithms cast in the maximum likelihood (ML) framework enable the use of all available statistical information---forms of diversity---in reality, they often deviate from their theoretical optimality properties due to improper estimation of the probability density function (PDF). This motivates the development of flexible ICA and IVA algorithms that closely adhere to the underlying statistical description of the data. Although it is attractive to let the data ''speak'' and hence minimize the assumptions, important prior information about the data, such as sparsity, is usually available. If incorporated into the ICA model, use of this additional information can relax the independence assumption, resulting in an improvement in the overall separation performance. Therefore, the development of a unified mathematical framework that can take into account both statistical independence and sparsity is of great interest.