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Modern random matrix theory (RMT) traces its origins to the fields of mathematics (through the works of Hurwitz) statistics (through Wishart) and of high energy physics (Wigner). We will give a brief historical survey of RMT and then proceed to the problem of deriving moments of functions of certain random matrices. Of particular interest are Wishart matrices and random matrices on the classical compact groups ( p) , ( p) and (2 p) . Some of the standard machinery for deriving moments will be reviewed and some open problems discussed, including the emergence of the Haar measure on ( ) p . Keywords: Random matrices, moments, Weingarten calculus, Haar measure.