The structure and functional properties of DNA derive from three classes of attributes with which the molecule is endowed. First, DNA within the cell nucleus is topologically constrained into loops by periodic attachments to a nuclear scaffold. This fixes the linking number Lk of each loop, the number of times either strand of the DNA double helix links through the closed loop formed by the other strand. The value of Lk is controlled by biological processes involving the introduction of transient strand breaks. Important biological activities (such as the initiation of gene expression) are modulated by this topological constraint. Second, DNA is not invariably a right-handed double helix, but can also assume other structures. These include left-handed helices, structures in which the two strands are separated, triple helices, etc. And, third, each of these conformations has an energy of formation, and effectively elastic mechanical properties - a lowest energy conformation, and an energy density function describing the energy required to deform the structure away from its lowest energy configuration. In general, the DNA structures that are possible, their unstressed conformations, and their hyperelastic energy densities all vary with the sequence of bases along the molecule. The interplay among these attributes provides a large variety of mathematical problems involving topologically constrained DNA that are central to many biological processes. This talk will survey a range of these problems, both solved and unsolved, and will suggest research opportunities in the field.