  We develop a systematic theory of eventually positive semigroups of linear
operators mainly on spaces of continuous functions. By eventually positive we
mean that for every positive initial condition the solution to the
corresponding Cauchy problem is positive for large enough time.
Characterisations of such semigroups are given by means of resolvent properties
of the generator and Perron--Frobenius type spectral conditions. We apply these
characterisations to prove eventual positivity of several examples of
semigroups including some generated by fourth order elliptic operators and a
delay differential equation. We also consider eventually positive semigroups on
arbitrary Banach lattices and establish several results for their spectral
bound which were previously only known for positive semigroups.
