  We review the recent developments in the theory of normal, normal self-dual
and general complex random matrices. The distribution and correlations of the
eigenvalues at large scales are investigated in the large $N$ limit. The 1/N
expansion of the free energy is also discussed. Our basic tool is a specific
Ward identity for correlation functions (the loop equation), which follows from
invariance of the partition function under reparametrizations of the complex
eigenvalues plane. The method for handling the loop equation requires the
technique of boundary value problems in two dimensions and elements of the
potential theory. As far as the physical significance of these models is
concerned, we discuss, in some detail, the recently revealed applications to
diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and
to the semiclassical behaviour of electronic blobs in the quantum Hall regime.
