The starting point is Lennart Carleson's celebrated theorem on the pointwise convergence. Over the last few years, this result, and its method of proof, has been the subject of renewed interest, and the result is the key result in an emerging set of Theorems.

We explain the Carleson theorem, why it is interesting, and elements of a new proof of the result obtained with Christoph Thiele. The new results related to Carleson's theorem shed light on the range of applicability of the ideas of the proof. A set of challenging problems remain open.