  Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum
Euclidean spaces -- are a standard example of a non-compact noncommutative
geometry. Recent progress in the harmonic analysis of these spaces gives us the
opportunity to highlight some of their peculiar features. For example, the
theory of nonlinear partial differential equations has unexpected properties in
this noncommutative setting. We develop elementary aspects of paradifferential
calculus for noncommutative Euclidean spaces and give some applications to
nonlinear evolution equations. We demonstrate how the analysis of some
equations radically simplifies in the strictly noncommutative setting.
