The chromosomes of eukaryotic cells are based on tremendously long DNA molecules that must be replicated and then physically separated to allow successful cell division. I will discuss what we have learned about chromosome structure from our group's biophysical experiments and mathematical modeling of chromosome structure. A key emerging feature of chromosome organization is the role of active chromatin loop formation, or "loop extrusion" as a mechanism leading to chromosome compaction, individualization, and segregation. I will discuss a number of aspects of the SMC complexes thought to be the loop-extruding elements. I will also discuss our group's studies of the role of chromosomal epigenetic marks in control of the structure and integrity of the cell nucleus.