  We consider simple random walks on Delaunay triangulations generated by point
processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes,
we show that the random walk satisfies an almost sure (or quenched) invariance
principle. This invariance principle holds for point processes which have
clustering or repulsiveness properties including Poisson point processes,
Mat{\'e}rn cluster and Mat{\'e}rn hardcore processes. The method relies on the
decomposition of the process into a martingale part and a corrector which is
proved to be negligible at the diffusive scale.
