In some sense deformation theory is as old as algebraic geometry itself: this is
because all algebro-geometric objects can be “deformed” by suitably varying
the coefficients of their defining equations, and this has of course always been
known by the classical geometers. Nevertheless a correct understanding of
what “deforming” means leads into the technically most difficult parts of
our discipline. It is fair to say that such technical obstacles have had a vast
impact on the crisis of the classical language and on the development of the
modern one, based on the theory of schemes and on cohomological methods.