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Spin glass models were introduced by physicists in the 1970s to model the statistical properties of certain magnetic materials. Over the last half century, these models have motivated a blossoming line of mathematical work with applications to multiple fields, at first sight distant from physics. This tutorial is deliberately written in a somewhat non-standard style, from several viewpoints. Rather than developing the theory in the most general setting, the authors focus on two concrete problems that are motivated by questions in statistical estimation. Their treatment is far from exhaustive, but they do not hesitate to pursue detours that are interesting, but indirectly related to the original questions posed by the examples. The authors also present a mixture of non-rigorous and rigorous techniques. The authors clearly indicate when something is proven and explain non-rigorous techniques on examples for which rigorous alternatives are available. Written by two recognized experts and based on a course given at Stanford University, this tutorial is a unique introduction to a topic that has many avenues for furthering research in statistics, mathematics, and computer science. It provides an accessible tutorial to understand and use the theories being deployed in physics for over 50 years.

This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. It is accessible to graduate students and researchers without a specific training in any of these fields. The selected topics include spin glasses, error correcting codes, satisfiability, and are central to each field. The approach focuses on large random instances and adopts a common probabilistic formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to study phase transitions.