Essential processes such as accurate chromosome segregation, regulation of gene expression and DNA repair rely on protein-mediated DNA tethering. Sister chromatid cohesion requires the SMC complex cohesin to act as a protein linker that holds replicated chromatids together (1, 2). The molecular mechanism by which cohesins hold sister chromatids has remained controversial. Here, we used a single molecule approach to visualise the activity of cohesin complexes as they hold DNA molecules. We describe a DNA bridging activity that requires ATP and is conserved from yeast to human cohesin. We show that cohesin can form two distinct classes of bridges at physiological conditions, a "permanent bridge" able to resists high force (over 80pN) and a "reversible bridge" that breaks at lower forces (5-40pN). Both classes of bridges require Scc2/Scc4 in addition to ATP. We demonstrate that bridge formation requires physical proximity of the DNA segments to be tethered and show that "permanent" cohesin bridges can move between two DNA molecules but cannot be removed from DNA when they occur in cis. This suggests that separate physical compartments in cohesin molecules are involved in the bridge. Finally, we show that cohesin tetramers, unlike condensin, cannot compact linear DNA molecules against low force, demonstrating that the core activity of cohesin tetramers is bridging DNA rather than compacting it. Our findings carry important implications for the understanding of the basic mechanisms behind cohesin-dependent establishment of sister chromatid cohesion and chromosome architecture.