The empirical spectral distribution of a non-selfadjoint random matrix concentrates around a deterministic probability measure on the complex plane as its dimension increases. Despite the inherent spectral instability of such models, this approximation is valid all the way down to local scales just above the typical eigenvalue spacing distance. We will present recent results on eigenvalue spectra for non-selfadjoint random matrices with correlated entries and their application to systems of randomly coupled differential equations that are used to model a wide range of disordered dynamical systems ranging from neural networks to food webs.

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