Light pseudo-Goldstone bosons 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 pseudo-Goldstone bosons 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.