  This monograph provides an overview on the Maurer-Cartan methods in algebra,
geometry, topology, and mathematical physics. It offers a conceptual,
exhaustive and gentle treatment of the twisting procedure, which functorially
creates new differential graded Lie algebras, associative algebras or operads
(as well as their homotopy versions) from a Maurer-Cartan element. The twisting
procedure for (homotopy) associative algebras or (homotopy) Lie algebras is
described by means of the action of the biggest deformation gauge group ever
considered. We give a criterion on quadratic operads for the existence of a
meaningful twisting procedure of their associated categories of algebras. And,
we introduce the twisting procedure for operads \`a la Willwacher using a new
and simpler presentation, which provides us with a wide source of motivating
examples related to graph homology, both recovering known graph complexes (due
to Kontsevich) and introducing some new ones. This book starts with elementary
surveys on gauge theory and deformation theory using differential graded Lie
algebras in order to ease the way to the theory. It finishes with concise
surveys on the fundamental theorem of deformation theory, higher Lie theory,
rational homotopy theory, simplicial theory of homotopy algebras, and the Floer
cohomology of Lagrangian submanifolds, to illustrate deep examples of
applications.
