  We discuss the problem of constructing models containing an axion and
axion-like particles, motivated by astrophysical observations, with decay
constants at the intermediate scale ranging from $10^9$GeV to $10^{13}$GeV. We
present examples in which the axion and axion-like particles arise accidentally
as pseudo Nambu-Goldstone bosons of automatic global chiral symmetries, in
models having exact discrete symmetries.
