I will describe a class of results on concentration in L_p Kantorovich distances of the empirical spectral measures of random matrices from various ensembles; in particular, I will discuss results for random matrices from the compact classical groups, various ensembles of Hermitian random matrices, the complex Ginibre ensemble, and a random matrix model of quantum spin chains. I will briefly sketch some of the proofs, which make use of some classical results on concentration of measure, entropy methods, and in some cases, the special structure of determinantal point processes.