Lovasz in the early seventies invented his celebrated local lemma (LLL) to prove the existence of combinatorial objects that satisfy a prescribed sparse set of constraints. His beautiful proof was inherently non-constructive. Nearly two decades later J. Beck presented a constructive proof, which was however seen as technical.

In 2008 Robin A. Moser and in 2009 Moser and Gabor Tardos turned the LLL research around by giving a constructive proof with a simple resample process (algorithm) at its heart.

We describe the original proof of Lovasz and the new proof Moser and Tardos. Then we discuss a great number of consequences of the original and new results.