  After introducing some motivations for this survey, we describe a formalism
to parametrize a wide class of algebraic structures occurring naturally in
various problems of topology, geometry and mathematical physics. This allows us
to define an "up to homotopy version" of algebraic structures which is coherent
(in the sense of $\infty$-category theory) at a high level of generality. To
understand the classification and deformation theory of these structures on a
given object, a relevant idea inspired by geometry is to gather them in a
moduli space with nice homotopical and geometric properties. Derived geometry
provides the appropriate framework to describe moduli spaces classifying
objects up to weak equivalences and encoding in a geometrically meaningful way
their deformation and obstruction theory. As an instance of the power of such
methods, I will describe several results of a joint work with Gregory Ginot
related to longstanding conjectures in deformation theory of bialgebras,
$E_n$-algebras and quantum group theory.
