  Toeplitz operators are met in different fields of mathematics such as
stochastic processes, signal theory, completeness problems, operator theory,
etc. In applications, spectral and mapping properties are of particular
interest. In this survey we will focus on kernels of Toeplitz operators. This
raises two questions. First, how can one decide whether such a kernel is non
trivial? We will discuss in some details the results starting with Makarov and
Poltoratski in 2005 and their succeeding authors concerning this topic. In
connection with these results we will also mention some intimately related
applications to completeness problems, spectral gap problems and P{\'o}lya
sequences. Second, if the kernel is non-trivial, what can be said about the
structure of the kernel, and what kind of information on the Toeplitz operator
can be deduced from its kernel? In this connection we will review a certain
number of results starting with work by Hayashi, Hitt and Sarason in the late
80's on the extremal function.
