Axions are light pseudoscalar bosons postulated with many motivations in particle physics and cosmology, including the strong CP problem and the dark matter in our Universe. In this lecture notes, we discuss
a variety of known ultraviolet (UV) theories for axions
and their low energy properties. We are primarily concerned with the
quantum chromodynamics axion solving the strong CP problem, as well as lighter axion-like particles.
In regard to their UV origin, such light axions may arise from the spontaneous breakdown of
a linearly realized global Peccei-Quinn $U(1)$ symmetry in the context of 4-dimensional effective field theories, or they may originate from a gauge field in higher dimensional theories.
It is noted that different UV models for these axions predict a distinctive pattern of low energy axion couplings, which may have interesting implications for laboratory, astrophysical, or cosmological studies of axions. We also provide an introductory discussion of the effective field theory for axions from
$p$-form gauge fields in string theory with concrete examples.