  We study Hilbert's fourteenth problem from a geometric point of view.
Nagata's celebrated counterexample demonstrates that for an arbitrary group
action on a variety the ring of invariant functions need not be isomorphic to
the ring of functions of an affine variety. Nevertheless one can prove that
such a ring of invariants is always isomorphic to the ring of functions on a
quasi-affine variety. Conversely, for a given quasi-affine variety V there
exists always an action of the additive group on some affine variety W such
that the ring of functions of V is isomorphic to the ring of invariant
functions on W. Thus a k-algebra occurs as invariant ring for some group acting
on a k-variety iff it occurs as function ring for some quasi-affine k-variety.
