  Building on the work of Crouseilles and Faou on the 2D case, we construct
$C^\infty$ quasi-periodic solutions to the incompressible Euler equations with
periodic boundary conditions in dimension 3 and in any even dimension. These
solutions are genuinely high-dimensional, which is particularly interesting
because there are extremely few examples of high-dimensional initial data for
which global solutions are known to exist. These quasi-periodic solutions can
be engineered so that they are dense on tori of arbitrary dimension embedded in
the space of solenoidal vector fields. Furthermore, in the two-dimensional case
we show that quasi-periodic solutions are dense in the phase space of the Euler
equations. More precisely, for any integer $N\geq 1$ we prove that any $L^q$
initial stream function can be approximated in $L^q$ (strongly when $1\leq q<
\infty$ and weak-* when $q=\infty$) by smooth initial data whose solutions are
dense on $N$-dimensional tori.
