  We prove that the supercritical phase of Voronoi percolation on
$\mathbb{R}^d$, $d\geq 3$, is well behaved in the sense that for every
$p>p_c(d)$ local uniqueness of macroscopic clusters happens with high
probability. As a consequence, truncated connection probabilities decay
exponentially fast and percolation happens on sufficiently thick 2D slabs. This
is the analogue of the celebrated result of Grimmett & Marstrand for Bernoulli
percolation and serves as the starting point for renormalization techniques
used to study several fine properties of the supercritical phase.
