  This thesis is divided into two parts. The first one is composed of
recollections on operad theory, model categories, simplicial homotopy theory,
rational homotopy theory, Maurer-Cartan spaces, and deformation theory. The
second part deals with the theory of convolution algebras and some of their
applications, as explained below.
  Suppose we are given a type of algebras, a type of coalgebras, and a
relationship between those types of algebraic structures (encoded by an operad,
a cooperad, and a twisting morphism respectively). Then, it is possible to
endow the space of linear maps from a coalgebra C and an algebra A with a
natural structure of Lie algebra up to homotopy. We call the resulting homotopy
Lie algebra the convolution algebra of A and C. We study the theory of
convolution algebras and their compatibility with the tools of homotopical
algebra: infinity morphisms and the homotopy transfer theorem. After doing
that, we apply this theory to various domains, such as derived deformation
theory and rational homotopy theory. In the first case, we use the tools we
developed to construct an universal Lie algebra representing the space of
Maurer-Cartan elements, a fundamental object of deformation theory. In the
second case, we generalize a result of Berglund on rational models for mapping
spaces between pointed topological spaces. In the last chapter of this thesis,
we give a new approach to two important theorems in deformation theory: the
Goldman-Millson theorem and the Dolgushev-Rogers theorem.
