  A famous theorem of Szemer\'edi asserts that any set of integers of positive
upper density will contain arbitrarily long arithmetic progressions. In its
full generality, we know of four types of arguments that can prove this
theorem: the original combinatorial (and graph-theoretical) approach of
Szemer\'edi, the ergodic theory approach of Furstenberg, the Fourier-analytic
approach of Gowers, and the hypergraph approach of Nagle-R\"odl-Schacht-Skokan
and Gowers. In this lecture series we introduce the first, second and fourth
approaches, though we will not delve into the full details of any of them. One
of the themes of these lectures is the strong similarity of ideas between these
approaches, despite the fact that they initially seem rather different.
