Axions have been originally introduced by R. Peccei and H. Quinn (PQ) as a possible solution of the
strong CP problem in QCD. Ever since, they have played a considerable role in particle physics and cosmology,
being one possible dark matter candidate in our universe. Their origin is related to an extension of the Standard Model
with a global anomalous (Peccei-Quinn) U(1) symmetry.
In the period 2005-2013 the PQ construction has been generalized to axions connected with abelian gauge anomalies,
both in non-supersymmetric and supersymmetric contexts.
In this first Leverhulme lecture we will describe the basic motivations of these models. Their close relatives, the dilatons, will be addressed in a second lecture.