  The harmonic formulation of Einstein's field equations is considered, where
the gauge conditions are introduced as dynamical constraints. The difference
between the fully constrained approach (used in analytical approximations) and
the free evolution one (used in most numerical approximations) is pointed out.
As a generalization, quasi-stationary gauge conditions are also discussed,
including numerical experiments with the gauge-waves testbed. The complementary
3+1 approach is also considered, where constraints are related instead with
energy and momentum first integrals and the gauge must be provided separately.
The relationship between the two formalisms is discussed in a more general
framework (Z4 formalism). Different strategies in black hole simulations follow
when introducing singularity avoidance as a requirement. More flexible
quasi-stationary gauge conditions are proposed in this context, which can be
seen as generalizations of the current 'freezing shift' prescriptions.
