  Spacetime is foliated by spatial hypersurfaces in the 3+1 split of General
Relativity. The initial value problem then consists of specifying initial data
for all relevant fields on one such a spatial hypersurface. These fields are
the 3-metric and extrinsic curvature together with matter fields such as fluid
velocity, energy density and rest mass density. There is a lot of freedom in
choosing such initial data. This freedom corresponds to the physical state of
the system at the initial time. At the same time the initial data have to
satisfy the Hamiltonian and momentum constraint equations of General Relativity
and can thus not be chosen completely freely. We discuss the conformal
transverse traceless and conformal thin sandwich decompositions that are
commonly used in the construction of constraint satisfying initial data. These
decompositions allow us to specify certain free data that describe the physical
nature of the system. The remaining metric fields are then determined by
solving elliptic equations derived from the constraint equations. We describe
initial data for single black holes and single neutron stars, and how we can
use conformal decompositions to construct initial data for binaries made up of
black holes or neutron stars. Orbiting binaries will emit gravitational
radiation and thus lose energy. Since the emitted radiation tends to
circularize the orbits over time, one can thus expect that the objects in a
typical binary move on almost circular orbits with slowly shrinking radii. This
leads us to the concept of quasi-equilibrium which essentially assumes that
time derivatives are negligible in corotating coordinates, for binaries on
almost circular orbits. We review how quasi-equilibrium assumptions can be used
to make physically well motivated approximations that simplify the elliptic
equations we have to solve.
