  An intriguing feature of positive $C_0$-semigroups on function spaces (or
more generally on Banach lattices) is that their long-time behaviour is much
easier to describe than it is for general semigroups. In particular, the
convergence of semigroup operators (strongly or in the operator norm) as time
tends to infinity can be characterized by a set of simple spectral and
compactness conditions.
  In the present paper, we show that similar theorems remain true for the
larger class of (uniformly) eventually positive semigroups - which recently
arose in the study of various concrete differential equations.
  A major step in one of our characterizations is to show a version of the
famous Niiro-Sawashima theorem for eventually positive operators. Several
proofs for positive operators and semigroups do not work in our setting any
longer, necessitating different arguments and giving our approach a distinct
flavour.
