  We develop a theory of eventually positive $C_0$-semigroups on Banach
lattices, that is, of semigroups for which, for every positive initial value,
the solution of the corresponding Cauchy problem becomes positive for large
times. We give characterisations of such semigroups by means of spectral and
resolvent properties of the corresponding generators, complementing existing
results on spaces of continuous functions. This enables us to treat a range of
new examples including the square of the Laplacian with Dirichlet boundary
conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neumann operator
on $L^2$ and the Laplacian with non-local boundary conditions on $L^2$ within
the one unified theory. We also introduce and analyse a weaker notion of
eventual positivity which we call "asymptotic positivity", where trajectories
associated with positive initial data converge to the positive cone in the
Banach lattice as $t \to \infty$. This allows us to discuss further examples
which do not fall within the above-mentioned framework, among them a network
flow with non-positive mass transition and a certain delay differential
equation.
