  Effective equations are often useful to extract physical information from
quantum theories without having to face all technical and conceptual
difficulties. One can then describe aspects of the quantum system by equations
of classical type, which correct the classical equations by modified
coefficients and higher derivative terms. In gravity, for instance, one expects
terms with higher powers of curvature. Such higher derivative formulations are
discussed here with an emphasis on the role of degrees of freedom and on
differences between Lagrangian and Hamiltonian treatments. A general scheme is
then provided which allows one to compute effective equations perturbatively in
a Hamiltonian formalism. Here, one can expand effective equations around any
quantum state and not just a perturbative vacuum. This is particularly useful
in situations of quantum gravity or cosmology where perturbations only around
vacuum states would be too restrictive. The discussion also demonstrates the
number of free parameters expected in effective equations, used to determine
the physical situation being approximated, as well as the role of classical
symmetries such as Lorentz transformation properties in effective equations. An
appendix collects information on effective correction terms expected from loop
quantum gravity and string theory.
