  We develop an approach to noncommutative algebraic geometry ``in the
perturbative regime" around ordinary commutative geometry. Let R be a
noncommutative algebra and A=R/[R,R] its commutativization. We describe what
should be the formal neighborhood of M=Spec(A) in the (nonexistent) space
Spec(R). This is a ringed space (M,O) where O is a certain sheaf of
noncommutative rings on M. Such ringed spaces can be glued together to form
more global objects called NC-schemes. We are especially interested in
NC-manifolds, NC-schemes for which the completion of O at every point of M is
isomorphic to the algebra of noncommutative power series (completion of the
free associative algebra). An explicit description of the simplest NC-manifold,
the affine space, is given by using the Feynman-Maslov calculus of ordered
operators. We show that many familiar algebraic varieties can be naturally
enlarged to NC-manifolds. Among these are all the classical flag varieties and
all the smooth moduli spaces of vector bundles.
