  This is the first paper of a five part work in which we study the Lagrangian
and Hamiltonian structure of classical field theories with constraints. Our
goal is to explore some of the connections between initial value constraints
and gauge transformations in such theories (either relativistic or not). To do
this, in the course of these four papers, we develop and use a number of tools
from symplectic and multisymplectic geometry. Of central importance in our
analysis is the notion of the ``energy-momentum map'' associated to the gauge
group of a given classical field theory. We hope to demonstrate that many
different and apparently unrelated facets of field theories can be thereby tied
together and understood in an essentially new way.
  In Part I we develop some of the basic theory of classical fields from a
spacetime covariant viewpoint. We begin with a study of the covariant
Lagrangian and Hamiltonian formalisms, on jet bundles and multisymplectic
manifolds, respectively. Then we discuss symmetries, conservation laws, and
Noether's theorem in terms of ``covariant momentum maps.''
