Graph complexes are differential graded vector spaces whose elements are linear combinations of combinatorial graphs with a differential given by some combinatorial rule such as contraction of edges. 
Many mathematical problems admit formulations in terms of graph complexes, in topics as distinct as knot theory, outer automorphisms of free groups and moduli spaces of curves.
In this talk, following the concrete problem of understanding the topology of configurations spaces of points I will give an introduction to the theory of graph complexes and show how these algebro-combinatorial tools can provide some problems with new insight.