  The theory of Fourier integral operators is surveyed, with an emphasis on
local smoothing estimates and their applications. After reviewing the classical
background, we describe some recent work of the authors which established sharp
local smoothing estimates for a natural class of Fourier integral operators. We
also show how local smoothing estimates imply oscillatory integral estimates
and obtain a maximal variant of an oscillatory integral estimate of Stein.
Together with an oscillatory integral counterexample of Bourgain, this shows
that our local smoothing estimates are sharp in odd spatial dimensions.
Motivated by related counterexamples, we formulate local smoothing conjectures
which take into account natural geometric assumptions arising from the
structure of the Fourier integrals.
