  We present the first examples of formally asymptotically flat black hole
solutions with horizons of general lens space topology $L(p,q)$. These
5-dimensional static/stationary spacetimes are regular on and outside the event
horizon for any choice of relatively prime integers $1\leq q<p$, in particular
conical singularities are absent. They are supported by Kaluza-Klein matter
fields arising from higher dimensional vacuum solutions through reduction on
tori. The technique is sufficiently robust that it leads to the explicit
construction of regular solutions, in any dimension, realising the full range
of possible topologies for the horizon as well as the domain of outer
communication, that are allowable with multi-axisymmetry. Lastly, as a by
product, we obtain new examples of regular gravitational instantons in higher
dimensions.
