  We obtain the mean field limit and the propagation of chaos for a system of
particles interacting with a singular interaction force of the type
$1/|x|^\alpha$, with $\alpha <1$ in dimension $d \geq 3$. We also provide
results for forces with singularity up to $\alpha < d-1$ but with large enough
cut-off. This last result thus almost includes the most interesting case of
Coulombian or gravitational interaction, but it is also interesting when the
strength of the singularity $\alpha$ is larger but close to one, in which case
it allows for very small cut-off.
