  Killing horizons which can be such for two or more linearly independent
Killing vectors are studied. We provide a rigorous definition and then show
that the set of Killing vectors sharing a Killing horizon is a Lie algebra
$\mathcal{A}_{\mathcal{H}}$ of dimension at most the dimension of the
spacetime. We prove that one cannot attach different surface gravities to such
multiple Killing horizons, as they have an essentially unique non-zero surface
gravity (or none). $\mathcal{A}_{\mathcal{H}}$ always contains an Abelian
(sub)-algebra ---whose elements all have vanishing surface gravity--- of
dimension equal to or one less than dim $\mathcal{A}_{\mathcal{H}}$. There
arise only two inequivalent possibilities, depending on whether or not there
exists the non-zero surface gravity. We show the connection with Near Horizon
geometries, and also present a linear system of PDEs, the master equation, for
the proportionality function on the horizon between two Killing vectors of a
multiple Killing horizon, with its integrability conditions. We provide
explicit examples of all possible types of multiple Killing horizons, as well
as a full classification of them in maximally symmetric spacetimes.
