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. We give sufficient criteria for local eventual positivity of the semigroup and apply them to concrete operators, for instance, the square of the Dirichlet Laplacian 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.