  We review recent efforts to re-formulate the Einstein equations for fully
relativistic numerical simulations. In order to complete a long-term and
accurate simulations of binary compact objects, people seek a robust set of
equations against the violation of constraints. Many trials have revealed that
mathematically equivalent sets of evolution equations show different numerical
stability in free evolution schemes. In this article, we overview the efforts
of the community, categorizing them into three directions: (1) modifications of
the standard Arnowitt-Deser-Misner equations initiated by the Kyoto group (the
so-called Baumgarte-Shapiro-Shibata-Nakamura equations), (2) rewriting the
evolution equations in a hyperbolic form, and (3) construction of an
"asymptotically constrained" system. We then introduce our series of works that
tries to explain these evolution behaviors in a unified way using eigenvalue
analysis of the constraint propagation equations. The modifications of (or
adjustments to) the evolution equations change the character of constraint
propagation, and several particular adjustments using constraints are expected
to damp the constraint-violating modes. We show several set of adjusted
ADM/BSSN equations, together with their numerical demonstrations.

General Relativity and Quantum Cosmology