We introduce the concept of topology weighting, a method for quantifying relationships between taxa that are not necessarily monophyletic, and visualizing how these relationships change across the genome. A given set of taxa can be related in a limited number of ways, but if each taxon is represented by multiple sequences, the number of possible topologies becomes very large. Topology weighting reduces this complexity by quantifying the contribution of each taxon topology to the full tree. We describe our method for topology weighting by iterative sampling of subtrees ($Twisst$), and test it on both simulated and real genomic data. Overall, we show that this is an informative and versatile approach, suitable for exploring relationships in almost any genomic dataset. Scripts to implement the method described are available at http://github.com/simonhmartin/twisst.