  A theorem of Pridham and Lurie provides an equivalence between formal moduli
problems and Lie algebras in characteristic zero. We prove a generalization of
this correspondence, relating formal moduli problems parametrized by algebras
over a Koszul operad to algebras over its Koszul dual operad. In particular,
when the Lie algebra associated to a deformation problem is induced from a
pre-Lie structure it corresponds to a permutative formal moduli problem. As
another example we obtain a correspondence between operadic formal moduli
problems and augmented operads.
