The study of small and local deformations of algebraic varieties originates in the classical work of Kodaira and Spencer and its formalization by Grothendieck in the late 1950's. It has become increasingly important in algebraic geometry in every context where variational phenomena come into play, and in classification theory. Today deformation theory is highly formalized and has ramified widely. This self-contained account of deformation theory in classical algebraic geometry (over an algebraically closed field) brings together for the first time some results previously scattered in the literature, with relatively little known proofs, yet of everyday relevance to algebraic geometers. It also includes applications to the construction and properties of Severi varieties of families of plane nodal curves, space curves, deformations of quotient singularities, Hilbert schemes of points, local Picard functors, etc. The exposition, amenable at graduate student level, includes many examples.