  The purpose of these notes is describe the state of progress on the
restriction problem in harmonic analysis, with an emphasis on the developments
of the past decade or so on the Euclidean space version of these problems for
spheres and other hypersurfaces. As the field is quite large and has so many
applications, it will be impossible to completely survey the field, but we will
try to at least give the main ideas and developments in this area.
  The restriction problem are connected to many other conjectures, notably the
Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as the
local smoothing conjecture. For reasons of space, we will not be able to
discuss all these connections in detail; our main focus will be on proving
restriction theorems for the Fourier transform.
