  Light pseudo-Nambu-Goldstone bosons (pNGBs) such as, e.g.~axion-like
particles, that are non-thermally produced via the misalignment mechanism are
promising dark matter candidates. An important feature of pNGBs is their
periodic potential, whose scale of periodicity controls all their couplings. As
a consequence of the periodicity the maximal potential energy is limited and,
hence, producing the observed dark matter density poses significant constraints
on the allowed masses and couplings. In the presence of a monodromy, the field
range as well as the range of the potential can be significantly extended. As
we argue in this paper this has important phenomenological consequences. The
constraints on the masses and couplings are ameliorated and couplings to
Standard Model particles could be significantly stronger, thereby opening up
considerable experimental opportunities. Yet, monodromy models can also give
rise to new and qualitatively different features. As a remnant of the
periodicity the potential can feature pronounced wiggles. When the field is
passing through them quantum fluctuations are enhanced and particles with
non-vanishing momentum are produced. Here, we perform a first analysis of this
effect and delineate under which circumstances this becomes important. We
discuss possible cosmological consequences.
