Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular values, delocalization of eigenvectors, and, among further applications, in establishing the limiting spectral distribution for certain random matrix models. Conversely, random linear operators play a very important role in high-dimensional convex geometry, as a tool in constructing special classes of convex sets and studying sections and projections of convex bodies. In this talk, I will discuss some recent results (by my collaborators and myself) on the borderline between convex geometry and the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to the study of the limiting spectral distribution), edges of the spectrum of sample covariance matrices, as well as some applications of random operators to questions in high-dimensional convex geometry.