From the back cover: Derived algebraic geometry is a powerful enhancement of algebraic geometry that has led to profound insights into various branches of mathematics. In this new formalism, some problems that were once either obscure or intractable now appear to become self-evident. In a first part of this volume, the reader will find standard prerequisites (schemes, infinity-categories, model categories, etc...), followed by a presentation of the foundations of derived algebraic geometry, as well as those of spectral algebraic geometry, its cousin theory, both providing a ＂spectralization＂ of algebraic geometry. The material is self-contained and is accessible to graduate students. The second part deals with applications of derived algebraic geometry (and spectral algebraic geometry) to a variety of problems: supersymmetric stacks, calculus of geometric stacks, higher Galois for Segal topoi, etc..., and most notably the mathematical models of reality. While this material builds up on the foundations given at the beginning, it is more accessible to researchers in algebraic geometry and physical mathematics. Graduate students will still find there new vistas for research.