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