Abstract: Random matrix models with complex non-Hermitian matrices are solved for finite matrix size N as well as in two different large-N limits at weak and strong non-Hermiticity. The tool we use are orthogonal polynomials in the complex plane, with Hermite or Laguerre polynomials and generalizations as examples. One can express the complex eigenvalue correlation functions or characteristic polynomials through these orthogonal polynomials and use their known asymptotics to take different large-N limits. New complex correlation functions are derived describing the fluctuations on the scale of the mean level spacing. We will also briefly discuss the issue of universality and possible physics applications.