  We describe a noncommutative deformation theory for presheaves and sheaves of
modules that generalizes the commutative deformation theory of these global
algebraic structures, and the noncommutative deformation theory of modules over
algebras due to Laudal.
  In the first part of the paper, we describe a noncommutative deformation
functor for presheaves of modules on a small category, and an obstruction
theory for this functor in terms of global Hochschild cohomology. An important
feature of this obstruction theory is that it can be computed in concrete terms
in many interesting cases.
  In the last part of the paper, we describe noncommutative deformation
functors for sheaves and quasi-coherent sheaves of modules on a ringed space
$(X, \mathcal{A})$. We show that for any good $\mathcal{A}$-affine open cover
$\mathsf{U}$ of $X$, the forgetful functor $\mathsf{QCoh}(\mathcal{A}) \to
\mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces an isomorphism of
noncommutative deformation functors.
  \emph{Applications.} We consider noncommutative deformations of
quasi-coherent $\mathcal{A}$-modules on $X$ when $(X, \mathcal{A}) = (X,
\mathcal{O}_X)$ is a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a
D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use
any open affine cover of $X$ closed under finite intersections to compute
noncommutative deformations in concrete terms using presheaf methods. We
compute the noncommutative deformations of the left $\mathcal{D}_X$-module
$\mathcal{O}_X$ when $X$ is an elliptic curve as an example.
