  This paper is a self-contained presentation of certain aspects of the theory
of weighted Sobolev spaces and elliptic operators on non-compact Riemannian
manifolds. Specifically, we discuss (i) the standard and weighted Sobolev
Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic
operators on manifolds with a finite number of ends modelled either on cones
("conifolds") or on cylinders. As an application of these results we present a
detailed analysis of certain spaces of harmonic functions on conifolds.
  Some of the results presented here are of course well-known. Some others are
probably known or self-evident to the experts. However, the current literature
is not always easy to understand and is often sketchy, apparently not covering
some aspects and consequences of the general theory which are useful in
applications. In particular, in recent years results of this type have played
an increasing role in Differential Geometry. The goal of this paper is thus to
fill certain gaps in the current literature and to make these results available
to a wider audience.
  The paper is also meant as a companion paper to the author's forthcoming
articles [8],[9]. From this point of view it is still incomplete: future
versions of this paper will incorporate more material, giving particular
attention to uniform elliptic estimates for certain parametric connect sum
constructions.
