A GinUE random matrix has all entries as independent standard complex Gaussians. The eigenvalue PDF can be explicitly calculated, revealing an analogy with the Boltzmann factor for a particular two-dimensional Coulomb gas. The functional form of the eigenvalue PDF leads to a determinantal structure for the general k-point correlation function. The corresponding correlation kernel admits bulk and edge scaling limits, while the density also admits a global scaling limit giving rise to an example of the circular law. Various generalisations of the GinUE permit analogous analysis. Those considered in the present chapter are the elliptic GinUE consisting of a linear combination of Hermitian and anti-Hermitian Gaussian random matrices; the induced GinUE which relates to a polar decomposition involving a rectangular generalisation of a GinUE matrix; the complex spherical ensemble of matrices \(G_1^{-1} G_2\) , with both \(G_1,G_2\) GinUE matrices; the sub-block of a Haar unitary random matrix; and products of both GinUE matrices, and of truncated Haar unitary matrices. In the original paper of Ginibre [293], the diagonalisation formula \(G = V \Lambda V^{-1}\) , where \(\Lambda \) is the diagonal matrix of eigenvalues, and V is the matrix of corresponding eigenvectors which are unique up to normalisation, was used as the starting point to derive the eigenvalue PDF ( 1.7 ).