  Black hole uniqueness is known to fail in higher dimensions, and the
multiplicity of black hole phases leads to phase transitions physics in General
Relativity. The black-hole black-string transition is a prime realization of
such a system and its phase diagram has been the subject of considerable study
in the last few years. The most surprising results seem to be the appearance of
critical dimensions where the qualitative behavior of the system changes, and a
novel kind of topology change. Recently, a full phase diagram was determined
numerically, confirming earlier predictions for a merger of the black-hole and
black string phases and giving very strong evidence that the end-state of the
Gregory-Laflamme instability is a black hole (in the dimension range 4<D<14).
Here this progress is reviewed, illustrated with figures, put into a wider
context, and the still open questions are listed.
