  The evolution of black-hole binaries in vacuum spacetimes constitutes the
two-body problem in general relativity. The solution of this problem in the
framework of the Einstein field equations is a substantially more complex
exercise than that of the dynamics of two point masses in Newtonian gravity,
but it also presents us with a wealth of new exciting physics. Numerical
methods are likely the only method to compute the dynamics of black-hole
systems in the fully non-linear regime and have been pursued since the 1960s,
culminating in dramatic breakthroughs in 2005. Here we review the methodology
and the developments that finally gave us a solution of this fundamental
problem of Einstein's theory and discuss the breakthrough's implication for the
wide range of contemporary black-hole physics.
