  We study Mandelbrot's percolation process in dimension $d \geq 2$. The
process generates random fractal sets by an iterative procedure which starts by
dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining
or discarding each subcube with probability $p$ or $1-p$ respectively. This
step is then repeated within the retained subcubes at all scales. As $p$ is
varied, there is a percolation phase transition in terms of paths for all $d
\geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$.
  For any $d \geq 2$, we consider the random fractal set produced at the
path-percolation critical value $p_c(N,d)$, and show that the probability that
it contains a path connecting two opposite faces of the cube $[0,1]^d$ tends to
one as $N \to \infty$. As an immediate consequence, we obtain that the above
probability has a discontinuity, as a function of $p$, at $p_c(N,d)$ for all
$N$ sufficiently large. This had previously been proved only for $d=2$ (for any
$N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation.
  In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N
\to \infty$, to the critical density $p_c$ of site percolation on the square
lattice. Assuming the existence of the correlation length exponent $\nu$ for
site percolation on the square lattice, we establish the speed of convergence
up to a logarithmic factor. In particular, our results imply that
$p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$, showing an
interesting relation with near-critical percolation.
