  We consider the question: ``How bad can the deformation space of an object
be?'' The answer seems to be: ``Unless there is some a priori reason otherwise,
the deformation space may be as bad as possible.'' We show this for a number of
important moduli spaces.
  More precisely, every singularity of finite type over Z (up to smooth
parameters) appears on: the Hilbert scheme of curves in projective space; and
the moduli spaces of smooth projective general-type surfaces (or
higher-dimensional varieties), plane curves with nodes and cusps, stable
sheaves, isolated threefold singularities, and more. The objects themselves are
not pathological, and are in fact as nice as can be: the curves are smooth, the
surfaces have very ample canonical bundle, the stable sheaves are torsion-free
of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies
Mumford's philosophy that even moduli spaces of well-behaved objects should be
arbitrarily bad unless there is an a priori reason otherwise.
  Thus one can construct a smooth curve in projective space whose deformation
space has any given number of components, each with any given singularity type,
with any given non-reduced behavior along various associated subschemes.
Similarly one can give a surface over F_p that lifts to Z/p^7 but not Z/p^8.
(Of course the results hold in the holomorphic category as well.)
  It is usually difficult to compute deformation spaces directly from
obstruction theories. We circumvent this by relating them to more tractable
deformation spaces via smooth morphisms. The essential starting point is Mnev's
Universality Theorem.
