  We develop a framework for derived deformation theory, valid in all
characteristics. This gives a model category reconciling local and global
approaches to derived moduli theory. In characteristic 0, we use this to show
that the homotopy categories of DGLAs and SHLAs (L infinity algebras)
considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible
with the derived stacks of Toen--Vezzosi and Lurie. Another application is that
the cohomology groups associated to any classical deformation problem (in any
characteristic) admit the same operations as Andre--Quillen cohomology.
