Axions are predicted to have no charge, very low mass (10-6 - 10-2 eV/c2) and very low interaction cross-sections for strong and weak forces. Hence they are nearly invisible to ordinary matter, and cannot be excluded on the basis of current measurements, though they have never been observed.
The predictions of axion theories would lead to them being created abundantly during the big bang. Because of a unique coupling to the instaton field of the primordial universe (i.e. "misalignment mechanism"), an effective dynamical friction is created during the acquisition of mass following cosmic inflation, this robs all such primordial axions of their kinetic energy. Hence axion theories predict that the universe will be filled with a very cold Bose-Einstein condensate of primordial axions. Depending on their mass, axions could plausibly explain the dark matter problem of cosmology. Observational studies to detect dark matter axions are underway, but they are not yet sufficiently sensitive to probe the mass regimes where axions would be expected to be found if they are the solution to the dark matter problem. Such studies have excluded the possibility of high mass axions.
It should be noted that the existence of axions are also a necessary component of string theory.