Much of random matrix theory is concerned with a relatively small collection of special random matrix models whose asymptotic behavior can be studied in stunning detail. But random matrices that arise naturally in other areas of pure and applied mathematics are often not of this form and may admit an essentially arbitrary structure. The main tools that have been used to study such models, known as matrix concentration inequalities, are extremely general but only provide crude information.

In the past few years, however, powerful new tools were discovered to obtain a sharp understanding, to leading order, of the spectral statistics of arbitrarily structured random matrices under surprisingly minimal assumptions. I will aim to describe how these methods work and explain how they enable the study of new applications, such as phase transition phenomena for nonhomogeneous random matrices, that are fundamentally outside the reach of traditional matrix concentration inequalities.

This talk is based on joint works with A. Bandeira, M. Boedihardjo, T. Brailovskaya, G. Cipolloni, and D. Schroeder.