We discuss results obtained jointly with Van Vu on the length of arithmetic progressions in
\(\ell \)-fold sumsets of the form
$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\} \end{aligned}$$
and
$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\text { all distinct}\}, \end{aligned}$$
where
\(\mathcal {A}\) is a set of integers. Applications are also discussed.