  This article is a survey on boundedness results for stable solutions to
semilinear elliptic problems. For these solutions, we present the currently
known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates
are known to hold up to dimension 4. They are expected to be true also in
dimensions 5 to 9, but this is still an open problem which has only been
established in the radial case.
