\indent Random matrix theory, a very young subject, studies the behaviour of the eigenvalues of matrices with random entries (with specified correlations). When all entries are independent (the simplest interesting assumption), a universal law emerges: essentially regardless of the laws of the entries, the eigenvalues become uniformly distributed in the unit disc as the matrix size increases. This {\em circular law} was first proved, with strong assumptions, in the 1980s; the current state of the art, due to Tao and Vu, with very weak assumptions, is less than a year old. It is the {\em universality} of the law that is of key interest. \\ \indent What if the entries are {\em not independent}? Of course, much more complex behaviour is possible in general. In the 1990s, ``$R$-diagonal'' matrix ensembles were introduced; they form a large class of non-normal random matrices with (typically) non-independent entries. In the last decade, they have found many uses in operator theory and free probability; most notably, they feature prominently in Haagerup's recent work towards proving the invariant subspace conjecture. \\ \indent In this lecture, I will discuss my recent joint work with Haagerup and Speicher, where we prove a universal law for the resolvent of any $R$-diagonal operator. The circular ensemble is an important special case. The rate of blow-up is, in fact universal among {\em all} $R$-diagonal operators, with a constant depending only on their fourth moment. The proof intertwines both complex analysis and combinatorics.\\ This talk will assume no knowledge of random matrix theory or free probability.