  Superintegrable systems in 2D Darboux spaces were classified and it was found
that there exist 12 distinct classes of superintegrable systems with quadratic
integrals of motion (and quadratic symmetry algebras generated by the
integrals) in the Darboux spaces. In this paper, we obtain exact solutions via
purely algebraic means for the energies of all the 12 existing classes of
superintegrable systems in four different 2D Darboux spaces. This is achieved
by constructing the deformed oscillator realization and finite-dimensional
irreducible representation of the underlying quadratic symmetry algebra
generated by quadratic integrals respectively for each of the 12
superintegrable systems. We also introduce generic cubic and quintic algebras,
generated respectively by linear and quadratic integrals and linear and cubic
integrals, and obtain their Casimir operators and deformed oscillator
realizations. As examples of applications, we present three classes of new
superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.
