  In this review we summarise recent results for the complex eigenvalues and
singular values of finite products of finite size random matrices, their
correlation functions and asymptotic limits. The matrices in the product are
taken from ensembles of independent real, complex, or quaternionic Ginibre
matrices, or truncated unitary matrices. Additional mixing within one ensemble
between matrices and their inverses is also covered. Exact determinantal and
Pfaffian expressions are given in terms of the respective kernels of orthogonal
polynomials or functions. Here we list all known cases and some straightforward
generalisations. The asymptotic results for large matrix size include new
microscopic universality classes at the origin and a generalisation of weak
non-unitarity close to the unit circle. So far in all other parts of the
spectrum the known standard universality classes have been identified. In the
limit of infinite products the Lyapunov and stability exponents share the same
normal distribution. To leading order they both follow a permanental point
processes. Our focus is on presenting recent developments in this rapidly
evolving area of research.
