Eigenvalues of GinSE matrices occur in complex conjugate pairs as for complex GinOE eigenvalues, but now with zero probability of an eigenvalue being real. Also in common with complex GinOE eigenvalues is that the GinSE eigenvalues form a Pfaffian point process, and that their PDF admits a Coulomb gas interpretation with image terms. While the determination of the required skew-orthogonal polynomials is straightforward, the particular form gives a new challenge in relation to analysing the various scaling limits. This is overcome by making use of a certain inhomogeneous partial differential equation satisfied by the pre-kernel. Such an approach carries over to the analysis of a weakly non-Hermitian regime. Also contained in this chapter is the large N analysis of the partition function and gap probability associated with GinSE eigenvalues, and results relating to the singular values and eigenvectors. Furthermore, a study is undertaken of an elliptic generalisation, which interpolates between GinSE and Hermitian GSE matrices.