A sunflower with r petals is a collection of r sets so that the
intersection of each pair is equal to the intersection of all. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w^w sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to c^w for some constant c. In this paper, we improve the bound to about (\log w)^w. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.