  The representation theory of deformed oscillator algebras, defined in terms
of an arbitrary function of the number operator~$N$, is developed in terms of
the eigenvalues of a Casimir operator~$C$. It is shown that according to the
nature of the $N$ spectrum, their unitary irreducible representations may fall
into one out of four classes, some of which contain bosonic, fermionic or
parafermionic Fock-space representations as special cases. The general theory
is illustrated by classifying the unitary irreducible representations of the
Arik-Coon, Chaturvedi-Srinivasan, and Tamm-Dancoff oscillator algebras, which
may be derived from the boson one by the recursive minimal-deformation
procedure of Katriel and Quesne. The effects on non-Fock-space representations
of the minimal deformation and of the quommutator-commutator transformation,
considered in such a procedure, are studied in detail.
