In the late 70's M. S. Livsic has discovered that a pair of commuting nonselfadjoint operators in a Hubert space, with finite nonhermitian ranks, satisfy a polynomial equation with constant (real) coefficients; in particular the joint spectrum of such a pair of operators lies on a certain algebraic curve in the complex plane, the so called discriminant curve of the pair of operators. More generally, it turns out that much in the same way as the study of a single nonself adjoint operator is intimately related to the function theory on the complex plane, more specifically on the upper half-plane, the study of a system of commuting nonself adjoint operators, at least with finite nonhermitian ranks, is related to the function theory on a compact Riemann surface of a higher genus, more specifically on a compact real Riemann surface. Prom a different perspective, while the study of a single nonselfadjoint operator leads to one-variable continuous time linear systems, the study of a pair of commuting nonselfadjoint operators leads to two-variable continuous time systems, which are necessarily overdetermined, hence must be considered together with an additional structure of compatibility conditions at the input and at the output. In this survey we give an introduction to the spectral theory of commuting nonselfadjoint operators and its interplay with system theory and the theory of Riemann surfaces and algebraic curves, including some recent results and open problems.