  The Willmore conjecture, proposed in 1965, concerns the quest to find the
best torus of all. This problem has inspired a lot of mathematics over the
years, helping bringing together ideas from subjects like conformal geometry,
partial differential equations, algebraic geometry and geometric measure
theory.
  In this article we survey the history of the conjecture and our recent
solution through the min-max approach. We finish with a discussion of some of
the many open questions that remain in the field.
