  We develop a framework for studying quasi-periodic maps and diffeomorphisms
on $\mathbb{R}^n$. As an application, we prove that the Euler equation is
locally well posed in a space of quasi-periodic vector fields on
$\mathbb{R}^n$. In particular, the equation preserves the spatial
quasi-periodicity of the initial data. Several results on the analytic
dependence of solutions on the time and the initial data are proved.
