  We give an exposition of the formal aspects of deformation theory in the
language of fibered categories, instead of the more traditional one of
functors. The main concepts are that of tangent space to a deformation problem,
obstruction theory, versal and universal formal deformations. We include proofs
of two key results: a versione of Schlessinger's Theorem in this context, and
the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a
detailed analysis of three important cases: smooth varieties, local complete
intersection subschemes and coherent sheaves.
