  We propose in these notes a list of some old and new questions related to
quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the
crucial influence of arithmetics on the dynamical features, with a strong
duality in general between Diophantine and Liouville behavior. We will discuss
rigidity and stability in Diophantine dynamics as well as their absence in
Liouville ones. Beyond this classical dichotomy between the Diophantine and the
Liouville worlds, we discuss some unified approaches and some phenomena that
are valid in both worlds. Our focus is mainly on low dimensional dynamics such
as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface
flows, as well as finite dimensional Hamiltonian systems.
  In an opposite direction, the study of the dynamical properties of some
diagonal and unipotent actions on the space of lattices can be applied to
arithmetics, namely to the theory of Diophantine approximations. We will
mention in the last section some problems related to that topic.
  The field of quasi-periodic dynamics is very extensive and has a wide range
of interactions with other mathematical domains. The list of questions we
propose is naturally far from exhaustive and our choice was often motivated by
our research involvements.
