  We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential
operators} with coefficients in general Hermitian vector bundles on Riemannian
manifolds, stressing a coordinate free approach that uses connections (which
are typically denoted $\nabla$). These concepts arise naturally from Partial
Differential Equations, including some that are formulated on plain Euclidean
domains, such as the weighted Sobolev spaces used to study PDEs on singular
domains. We prove several basic properties of the $\nabla$-Sobolev spaces and
of the $\nabla$-differential operators on general manifolds. For instance, we
prove mapping properties for our differential operators and independence of the
$\nabla$-Sobolev spaces on the choices of the connection $\nabla$ with respect
to totally bounded perturbations. We introduce a {\em Fr\'echet finiteness
condition} (FFC) for totally bounded vector fields, which is satisfied, for
instance, by open subsets of manifolds with bounded geometry. When (FFC) is
satisfied, we provide several equivalent definitions of our $\nabla$-Sobolev
spaces and of our $\nabla$-differential operators. We examine in more detail
the particular case of domains in the Euclidean space, including the case of
weighted Sobolev spaces. We also introduce and study the notion of a {\em
$\nabla$-bidifferential} operator (a bilinear version of differential
operators), obtaining results similar to those obtained for
$\nabla$-differential operators. Bilinear differential operators are necessary
for a global, geometric discussion of variational problems. We tried to write
the paper so that it is accessible to a large audience.
