The thermodynamics of spin glasses, and more generally the statistical mechanics of quenched disorder, is a problem of general interest to physicists and mathematicians from multiple disciplines and backgrounds. Its importance was recognized in 2021 with the awarding of the Nobel Prize in Physics to Giorgio Paris for his replica symmetry breaking solution of the Sherrington-Kirkpatrick model (a mean-field model of spin glasses) and its application to other problems in complex systems.

I will begin with a brief survey of spin glasses from both experimental and theoretical perspectives, followed by a description of some of the main thermodynamic features of Parisi's replica symmetry breaking solution of the mean-field spin glass. I will then turn to the problem of understanding the nature of the spin glass phase in nearest-neighbor spin glass models in finite dimensions. The central question to be addressed is the nature of broken symmetry in these systems. This is still a subject of controversy, and although the issues surrounding it have become more sharply defined in recent years, it remains an open question. I will explore this problem and present a unified picture of our current understanding of the structure of the spin glass phase in finite dimensions.