Abstract: Sub-Riemannian geometry is, roughly speaking, differential geometry relative to a singular, or degenerate, metric. For those of us working in nonsmooth geometric analysis, sub-Riemannian spaces provide a rich source of examples motivating a more general, abstract theory. I will describe the geometric and analytic structure of the simplest non-Euclidean sub-Riemannian space -- the three-dimensional Heisenberg group -- and its connection with the classical isoperimetric problem of Queen Dido (discussed in Virgil's Aeneid). The sub-Riemannian isoperimetric problem in the Heisenberg group (among all sets in the Heisenberg group with fixed volume, classify those which minimize the sub-Riemannian surface area) is a famous and long-standing open question. I'll trace the history of this problem and describe its current status. I'll also discuss some related analytic inequalities (Sobolev-Poincare inequalities) in this setting.