  The gravitational couplings of matter are studied in the presence of Lorentz
and CPT violation. At leading order in the coefficients for Lorentz violation,
the relativistic quantum hamiltonian is derived from the gravitationally
coupled minimal Standard-Model Extension. For spin-independent effects, the
nonrelativistic quantum hamiltonian and the classical dynamics for test and
source bodies are obtained. A systematic perturbative method is developed to
treat small metric and coefficient fluctuations about a Lorentz-violating and
Minkowski background. The post-newtonian metric and the trajectory of a test
body freely falling under gravity in the presence of Lorentz violation are
established. An illustrative example is presented for a bumblebee model. The
general methodology is used to identify observable signals of Lorentz and CPT
violation in a variety of gravitational experiments and observations, including
gravimeter measurements, laboratory and satellite tests of the weak equivalence
principle, antimatter studies, solar-system observations, and investigations of
the gravitational properties of light. Numerous sensitivities to coefficients
for Lorentz violation can be achieved in existing or near-future experiments at
the level of parts in 10^3 down to parts in 10^{15}. Certain coefficients are
uniquely detectable in gravitational searches and remain unmeasured to date.
