abstract: 1 Homology of local systems and characteristic varieties 2 Homology of free abelian covers 3 Resonance varieties, tangent cones, formality 4 Bieri-Neumann-Strebel-Renz invariants 5 Applications: a Knots, links, and 3-dimensional manifolds b Right-angled Artin groups and toric complexes c Kahler and quasi-Kahler manifolds d Hyperplane arrangements