  We begin a systematic study of Quantum Energy Inequalities (QEIs) in relation
to local covariance. We define notions of locally covariant QEIs of both
'absolute' and 'difference' types and show that existing QEIs satisfy these
conditions. Local covariance permits us to place constraints on the
renormalised stress-energy tensor in one spacetime using QEIs derived in
another, in subregions where the two spacetimes are isometric. This is of
particular utility where one of the two spacetimes exhibits a high degree of
symmetry and the QEIs are available in simple closed form. Various general
applications are presented, including a priori constraints (depending only on
geometric quantities) on the ground-state energy density in a static spacetime
containing locally Minkowskian regions. In addition, we present a number of
concrete calculations in both two and four dimensions which demonstrate the
consistency of our bounds with various known ground- and thermal state energy
densities. Examples considered include the Rindler and Misner spacetimes, and
spacetimes with toroidal spatial sections. In this paper we confine the
discussion to globally hyperbolic spacetimes; subsequent papers will also
discuss spacetimes with boundary and other related issues.
