‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group U _n , O _n , or USp _n , giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.