  This is the first one of a series of papers about zeta regularization of the
divergences appearing in the vacuum expectation value (VEV) of several local
and global observables in quantum field theory. More precisely we consider a
quantized, neutral scalar field on a domain in any spatial dimension, with
arbitrary boundary conditions and, possibly, in presence of an external
classical potential. We analyze, in particular, the VEV of the stress-energy
tensor, the corresponding boundary forces and the total energy, thus taking
into account both local and global aspects of the Casimir effect. In comparison
with the wide existing literature on these subjects, we try to develop a more
systematic approach, allowing to treat specific configurations by mere
application of a general machinery. The present Part I is mainly devoted to
setting up this general framework; at the end of the paper, this is exemplified
in a very simple case. In Parts II, III and IV we will consider more engaging
applications, indicated in the Introduction of the present work.
