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The Ginibre unitary ensemble (GinUE) consists of $N \times N$ random matriceswith independent complex standard Gaussian entries. This was introduced in 1965by Ginbre, who showed that the eigenvalues form a determinantal point processwith an explicit correlation kernel, and after scaling they are supported onthe unit disk with constant density. For some time now it has been appreciatedthat GinUE has a fundamental place within random matrix theory, both for itsapplications and for the richness of its theory. Here we review the progress ona number of themes relating to the study of GinUE. These are eigenvalueprobability density functions and correlation functions, fluctuation formulas,sum rules and asymptotic behaviours of correlation functions, and normal matrixmodels. We discuss too applications in quantum many body physics and quantumchaos, and give an account of some statistical properties of the eigenvectors.

Progress on the study of the Ginibre ensembles I: GinUE