  Scenarios with axion-like particles of variable masses, from the milli-eV
range to ultralight, can be generated in a natural way in the context of
effective field theory models with anomalous $U(1)$ gauge symmetries. They
include dimension-5 operators which define a coupling of the axion to the gauge
anomaly. They provide a realization, in the domain of ordinary gauge theories,
of models for such particles, which evade the usual mass/coupling constraint of
ordinary (Peccei-Quinn) axions and are natural dark matter candidates. As an
illustration of these models, we present an overview of two of these scenarios.
One of them is built around the next-to-minimal MSSM (NMSSM), a model called
the USSM-A, which illustrates how the St\"uckelberg supermultiplet can be used
to generate two dark matter candidates, a neutralino containing an axino
component coming from the St\"uckelberg sector, plus the axion ($\textrm{Im}\,
b$). The real component of the complex St\"uckelberg field carries dilaton-like
($\textrm{Re}\, b\, FF$) interactions. In a second model, non-supersymmetric,
the St\"uckelberg scale is raised up to the GUT epoch. In this case the axion
mass can be ultralight ($\sim 10^{-20}$ eV). The periodic potential generated
at the GUT phase transition and the corresponding oscillations are related to a
particle whose De Broglie wavelength can be sub-galactic. A similar analysis is
also possible for the supersymmetric scenario.
