  We generalize the notion of semi-universality in the classical deformation
problems to the context of derived deformation theories. A criterion for a
formal moduli problem to be semi-prorepresentable is produced. This can be seen
as an analogue of Schlessinger's conditions for a functor of Artinian rings to
have a semi-universal element. We also give a sufficient condition for a
semi-prorepresentable formal moduli problem to admit a $G$-equivariant
structure in a sense specified below, where $G$ is a linearly reductive group.
Finally, by making use of these criteria, we derive many classical results
including the existence of ($G$-equivariant) formal semi-universal deformations
of algebraic schemes and that of complex compact manifolds.
