The Hilbert matrix is an infinite matrix that is very simple to describe. It is a prototype of a Hankel operator on the space of square summable sequences, where its spectrum is well understood. Various related results were obtained in the 1950s.

The Hilbert matrix also induces a bounded operator on various other sequence spaces as well as on different spaces of analytic functions (including certain Hardy

and Bergman spaces), as was noticed from 2000 on. We review these more recent developments, including norm computations on different spaces and some most

recent results regarding the spectrum of the induced operator, covering also our recent work in progress with Aleman and Siskakis.