  We initiate a theory of locally eventually positive operator semigroups on
Banach lattices. Intuitively this means: given a positive initial datum, the
solution of the corresponding Cauchy problem becomes (and stays) positive in a
part of the domain, after a sufficiently large time. A drawback of the present
theory of eventually positive $C_0$-semigroups is that it is applicable only
when the leading eigenvalue of the semigroup generator has a strongly positive
eigenvector. We weaken this requirement and give sufficient criteria for
individual and uniform local eventual positivity of the semigroup. This allows
us to treat a larger class of examples by giving us more freedom on the domain
when dealing with function spaces -- for instance, the square of the Laplace
operator with Dirichlet boundary conditions on $L^2$ and the Dirichlet
bi-Laplacian on $L^p$-spaces. Besides, we establish various spectral and
convergence properties of locally eventually positive semigroups.
