  We consider the non-local Fisher-KPP equation modeling a population with
individuals competing with each other for resources with a strength related to
their distance, and obtain the asymptotics for the position of the invasion
front starting from a localized population. Depending on the behavior of the
competition kernel at infinity, the location of the front is either $2t -
({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some
explicit $\beta \in (0,1)$. Our main tools here are alocal-in-time Harnack
inequality and an analysis of the linearized problem with a suitable moving
Dirichlet boundary condition. Our analysis also yields, for any
$\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f\_\beta$ such
that the front for the localFisher-KPP equation with reaction term $f\_\beta$
is at $2t - O(t^\beta)$.
