Some spectral properties of products of random matrices are discussed.  In particular, exact analytic expression related to the eigenvalue and singular value distributions for the product of an arbitrary number of independent rectangular Gaussian random matrices are derived in the limit of large matrix dimensions.  The eigenvalue density is obtained by means of a planar diagrammatic technique, which will be outlined in great detail throughout the talk, while the singular value density is derived in the framework of Free Probability, the extension of ordinary probability theory to non-commuting objects.  Both distributions are shown to display a power-law behavior at zero. A heuristic form for finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions is proposed.  All results are compared to empirical spectra, finding an excellent overall agreement between theory and data.  Also, some possible applications to information theory and to problems of multivariate statistics are discussed.