  The main purpose of this article is to develop an explicit derived
deformation theory of algebraic structures at a high level of generality,
encompassing in a common framework various kinds of algebras (associative,
commutative, Poisson...) or bialgebras (associative and coassociative, Lie,
Frobenius...), that is algebraic structures parametrized by props. A central
aspect is that we define and study moduli spaces of deformations of algebraic
structures up to quasi-isomorphisms (and not just isotopies or isomorphisms).
To do so, we implement methods coming from derived algebraic geometry, by
encapsulating these deformation theories as classifying (pre)stacks with good
infinitesimal properties and derived formal geometry, by means of derived
formal moduli problems and derived formal groups. In particular, we prove that
the Lie algebra describing the deformation theory of an object in a given
$\infty$-category of dg algebras can be obtained equivalently as the tangent
complex of loops on a derived quotient of this moduli space by the homotopy
automorphims of this object. Moreover, we provide explicit formulae for such
derived deformation problems of algebraic structures up to quasi-isomorphisms
and relate them in a precise way to other standard deformation problems of
algebraic structures. This relation is given by a fiber sequence of the
associated dg-Lie algebras of their deformation complexes. Our results provide
simultaneously a vast generalization of standard deformation theory of
algebraic structures which is suitable (and needed) to set up algebraic
deformation theory both at the $\infty$-categorical level and at a higher level
of generality than algebras over operads. In addition, we study a general
criterion to compare formal moduli problems of different algebraic structures
and apply our formalism to $E_n$-algebras and bialgebras.
