Mammalian genomes are spatially organized into compartments, topologically associating domains (TADs), and loops to facilitate gene regulation and other chromosomal functions. How compartments, TADs, and loops are generated is unknown. It has been proposed that cohesin forms TADs and loops by extruding chromatin loops until it encounters CTCF, but direct evidence for this hypothesis is missing. Here, we show that cohesin suppresses compartments but is required for TADs and loops, that CTCF defines their boundaries, and that the cohesin unloading factor WAPL and its PDS5 binding partners control the length of loops. In the absence of WAPL and PDS5 proteins, cohesin forms extended loops, presumably by passing CTCF sites, accumulates in axial chromosomal positions (vermicelli), and condenses chromosomes. Unexpectedly, PDS5 proteins are also required for boundary function. These results show that cohesin has an essential genome‐wide function in mediating long‐range chromatin interactions and support the hypothesis that cohesin creates these by loop extrusion, until it is delayed by CTCF in a manner dependent on PDS5 proteins, or until it is released from DNA by WAPL. Synopsis Spatial organization of mammalian genomes facilitates gene regulation and replication. Hi‐C data reveal that cohesin binding at CTCF sites controls long‐range chromatin interactions, supporting a model in which chromatin loop extrusion drives genome 3D architecture. Cohesin suppresses compartments but is required for TADs and loops. CTCF and PDS5 proteins define TAD boundaries and loop anchors. WAPL and PDS5 proteins define the length of loops. Cohesin appears to form long‐range contacts by loop extrusion. Hi‐C data show that cohesin is required for long‐range chromatin interactions, supporting a model of loop extrusion driving the spatial organization of mammalian genomes.