  The first part of this article is a brief survey of the properties of
so-called almost interior points in ordered Banach spaces. Those vectors can be
seen as a generalization of ``functions which are strictly positive almost
everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach
lattices.
  In the second part we study the long--term behaviour of strongly positive
operator semigroups on ordered Banach spaces; these are semigroups which, in a
sense, map every non-zero positive vector to an almost interior point. Using
the Jacobs--de Leeuw--Glicksberg decomposition together with the theory
presented in the first part of the paper we deduce sufficiency criteria for
such semigroups to converge (strongly or in operator norm) as time tends to
infinity. This generalises known results for semigroups on Banach lattices as
well as on normally ordered Banach spaces with unit.
