How likely is a random matrix to be invertible? This fundamental question is intimately related to the question of estimating the smallest singular value of the random matrix. We discuss sharp estimates for the smallest singular value in the case of two novel ensembles of random matrices: inhomogeneous random matrices (whose entries are independent, but the variance profile is fairly general), and log-concave isotropic random matrices. When it comes to the latter, we will witness an exciting general phenomenon: convexity can replace independence in the study of universality in high dimensions. One important tool that we develop is an efficient discretization procedure of the sphere in high dimensions.We will flesh out the entertaining proof of this result. The talk will be based on three papers, two of which are joint with others: one with Tikhomirov and Vershynin, and another with Fernandez and Mui.