  This paper presents a survey on formal moduli problems. It starts with an
introduction to pointed formal moduli problems and a sketch of proof of a
Theorem (independently proven by Lurie and Pridham) which gives a precise
mathematical formulation for Drinfeld's derived deformation theory philosophy,
which gives a correspondence between formal moduli problems and differential
graded Lie algebras. The second part deals with Lurie's general theory of
deformation contexts, which we present in a slightly different way than the
original paper, emphasising the (more symmetric) notion of Koszul duality
contexts and morphisms thereof. In the third part, we explain how to apply this
machinery to the case of non-split formal moduli problems under a given derived
affine scheme; this situation has been dealt with recently by Joost Nuiten, and
requires to replace differential graded Lie algebras with differential graded
Lie algebroids. In the last part, we globalize this to the more general setting
of formal thickenings of derived stacks, and suggest an alternative approach to
results of Gaitsgory and Rozenblyum.
