  Fractal percolation has been introduced by Mandelbrot in 1974. We study the
two-dimensional case, with integer subdivision index M and survival probability
p. It is well known that there exists a non-trivial critical value p_c(M) such
that a.s. the largest connected component in the limiting set K is a point for
p<p_c(M) and with positive probability there is a connected component
intersecting opposite sides of the unit square for p\geq p_c(M).
  For all M\geq 2, the value of p_c(M) is unknown. In this paper we present
ideas to find lower and upper bounds, significantly sharper than those already
known. To find lower bounds, we compare fractal percolation with site
percolation. A fundamentally new result is that for all M we construct an
increasing sequence that converges to p_c(M). The terms in the sequence can in
principle be calculated algorithmically. These ideas lead to (computer aided)
proofs that p_c(2)> 0.881 and p_c(3)>0.784.
  For the upper bounds, we introduce the idea of classifications. The fractal
percolation iteration process now induces an iterative random process on a
finite alphabet, which is easier to analyze than the original process. This
theoretical framework is the basis of computer aided proofs for the following
upper bounds: p_c(2)<0.993, p_c(3)<0.940 and p_c(4)<0.972.
