Koszul modules and their associated resonance varieties are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. I will offer an introduction to this circle of ideas and discuss a recent proof of the Chen Ranks Conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. Applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of the arrangement will be presented. This is joint work with Aprodu, Raicu and Suciu.
