Motivated by growing interest in non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, I will discuss a few results on recently identified universality class AI^{\dagger} (complex symmetric matrices) as well as on a more standard complex Ginibre matrices (class A).
In the former class the exact finite-N mean eigenvalue density will be discussed together with its bulk and edge limits, the edge behaviour conjectured to be a characteristic feature of different non-Hermitian universality classes. In the Ginibre case I will present an explicit expression for the parametric covariance of spectral densities at the microscopic scale and ensuing parametric number variance in the bulk of the spectrum. A relation between parametric correlations and the distribution of the eigenvector non-orthogonality factor, which attracted considerable interest in recent years, will be demonstrated. The first part of the talk will be based on a joint work arXiv:2511.21643 with Gernot Akemann and Dmitry Savin and second part on a joint work in progress with Bertrand Lacroix-A-Chez-Toine.