Topology is the study of geometric properties that are preserved during bending, twisting and stretching of objects. In the context of the genome, topology is discussed at two interconnected and overlapping levels. The first focuses the DNA double helix itself, and includes alterations such as those triggered by DNA interacting proteins, processes which require the separation of the two DNA strands and DNA knotting. The second level is centered on the higher order organization of DNA into chromosomes, as well as dynamic conformational changes that occur on a chromosomal scale. Here, we refer to the first level as "DNA topology", the second as "chromosome topology". Since their identification, evidences suggesting that the so called structural maintenance of chromosomes (SMC) protein complexes are central to the interplay between DNA and chromosome topology have accumulated. The SMC complexes regulate replication, segregation, repair and transcription, all processes which influence, and are influenced by, DNA and chromosome topology. This review focuses on the details of the relationship between the SMC complexes and topology. It also discusses the possibility that the SMC complexes are united by a capability to sense the geometrical chirality of DNA crossings.