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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 Stückelberg supermultiplet can be used to generate two dark matter candidates, a neutralino containing an axino component coming from the Stückelberg sector, plus the axion ($\textrm{Im}\, b$). The real component of the complex Stückelberg field carries dilaton-like ($\textrm{Re}\, b\, FF$) interactions. In a second model, non-supersymmetric, the Stückelberg 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.