Leray complexes are, roughly speaking, simplicial complexes of bounded hereditary homological dimension, and they arise naturally in the study of topological extensions of Helly's theorem and its relatives. We'll describe some old and new results on Leray complexes with geometrical and combinatorial applications. In particular we'll discuss
1) A topological extension of the Lovász colorful Helly theorem (conjectured by Bárány).
2) Relations with commutative algebra and a proof of a conjecture of Terai on monomial ideals.
3) Covering vs. fractional covering in hypergraphs with bounded Leray numbers.
(Joint work with Gil Kalai.)