We study the realisation of axion inflation models in the complex structure moduli spaces of Calabi-Yau threefolds and fourfolds. The axions arise close to special points of these moduli spaces that admit discrete monodromy symmetries of infinite order. Examples include the large complex structure point and conifold point, but can be of more general nature. In Type IIB and F-theory compactifications the geometric axions receive a scalar potential from a flux-induced superpotential. We find toy variants of various inflationary potentials including the ones for natural inflation of one or multiple axions, or axion monodromy inflation with polynomial potential. Interesting examples are also given by mirror geometries of torus fibrations with Mordell-Weil group of rank $N-1$ or an $N$-section, which admit an axion if $N>3$.

41 pages, 3 figures