“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.”