We review the latest progress in understanding the phase structure of static
and neutral Kaluza-Klein black holes, i.e. static and neutral solutions of pure
gravity with an event horizon that asymptote to a d-dimensional Minkowski-space
times a circle. We start by reviewing the (mu,n) phase diagram and the split-up
of the phase structure into solutions with an internal SO(d-1) symmetry and
solutions with Kaluza-Klein bubbles. We then discuss the uniform black string,
non-uniform black string and localized black hole phases, and how those three
phases are connected, involving issues such as classical instability and
horizon-topology changing transitions. Finally, we review the bubble-black hole
sequences, their place in the phase structure and interesting aspects such as
the continuously infinite non-uniqueness of solutions for a given mass and
relative tension.Comment: 23 pages, 5 figures. v2: Typo fixe