Conjugation invariant ensembles of random matrices have long formed one of the basic paradigms in Random Matrix Theory. Apart from the Gaussian case, the matrix elements of a conjugation invariant random matrix are highly correlated, and this fact has traditionally been viewed as prohibiting the use of moment methods in the spectral analysis of invariant ensembles. However, it turns out that there is a very natural and appealing version of the moment method available for these ensembles which seems to have been overlooked. I will describe the rudiments of this method, and some of its applications. Based on joint work with Sho Matsumoto.