  This paper provides a conceptual study of the twisting procedure, which
amounts to create functorially new differential graded Lie algebras,
associative algebras or operads (as well as their homotopy versions) from a
Maurer--Cartan element. On the way, we settle the integration theory of
complete pre-Lie algebras in order to describe this twisting procedure in terms
of gauge group action. We give a criterion on quadratic operads for the
existence of a meaningful twisting procedure of their associated categories of
(homotopy) algebras. We also give a new presentation of the twisting procedure
for operads \`a la Willwacher and we perform new homology computations of graph
complexes.
