The need for a noncommutative algebraic geometry is apparent in
classical invariant and moduli theory. It is, in general, impossible
to find commuting parameters parametrizing all orbits of a Lie group
acting on a scheme. When one orbit is contained in the closure of
another, the orbit space cannot, in a natural way, be given a scheme
structure.
In this paper we shall show that one may overcome these
difficulties by introducing a noncommutative algebraic geometry,
where affine "schemes" are modeled on associative algebras. The
points of such an affine scheme are the simple modules of the
algebra, and the local structure of the scheme at a finite family
of points, is expressed in terms of a noncommutative deformation
theory proposed by the author in \cite{Laudal2002}.
More generally, the geometry in the theory is represented by a
{\it swarm}, i.e. a diagram (finite or infinite) of objects (and
if one wants, arrows) in a given $k$-linear Abelian category ($k$
a field), satisfying some reasonable conditions. The
noncommutative deformation theory refered to above, permits the
construction of a presheaf of associative $k$-algebras, locally
{\it parametrizing} the diagram. It is shown that this theory, in
a natural way, generalizes the classical scheme theory. Moreover
it provides a promising framework for treating problems of
invariant theory and moduli problems. In particular it is shown
that many moduli spaces in classical algebraic geometry are
commutativizations of noncommutative schemes containing additional
information.