  The thermal or equilibrium ensemble is one of the most ubiquitous states of
matter. For models comprised of many locally interacting quantum particles, it
describes a wide range of physical situations, relevant to condensed matter
physics, high energy physics, quantum chemistry and quantum computing, among
others. We give a pedagogical overview of some of the most important universal
features about the physics and complexity of these states, which have the
locality of the Hamiltonian at its core. We focus on mathematically rigorous
statements, many of them inspired by ideas and tools from quantum information
theory. These include bounds on their correlations, the form of the subsystems,
various statistical properties, and the performance of classical and quantum
algorithms. We also include a summary of a few of the most important technical
tools, as well as some self-contained proofs.
