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In this review we summarise recent results for the complex eigenvalues andsingular values of finite products of finite size random matrices, theircorrelation functions and asymptotic limits. The matrices in the product aretaken from ensembles of independent real, complex, or quaternionic Ginibrematrices, or truncated unitary matrices. Additional mixing within one ensemblebetween matrices and their inverses is also covered. Exact determinantal andPfaffian expressions are given in terms of the respective kernels of orthogonalpolynomials or functions. Here we list all known cases and some straightforwardgeneralisations. The asymptotic results for large matrix size include newmicroscopic universality classes at the origin and a generalisation of weaknon-unitarity close to the unit circle. So far in all other parts of thespectrum the known standard universality classes have been identified. In thelimit of infinite products the Lyapunov and stability exponents share the samenormal distribution. To leading order they both follow a permanental pointprocesses. Our focus is on presenting recent developments in this rapidlyevolving area of research.

Recent exact and asymptotic results for products of independent random matrices