  We study general aspects of the reductive dual pair correspondence, also
known as Howe duality. We make an explicit and systematic treatment, where we
first derive the oscillator realizations of all irreducible dual pairs:
$(GL(M,\mathbb R), GL(N,\mathbb R))$, $(GL(M,\mathbb C), GL(N,\mathbb C))$,
$(U^*(2M), U^*(2N))$, $(U(M_+,M_-), U(N_+,N_-))$, $(O(N_+,N_-),Sp(2M,\mathbb
R))$, $(O(N,\mathbb C), Sp(2M,\mathbb C))$ and $(O^*(2N), Sp(M_+,M_-))$. Then,
we decompose the Fock space into irreducible representations of each group in
the dual pairs for the cases where one member of the pair is compact as well as
the first non-trivial cases of where it is non-compact. We discuss the
relevance of these representations in several physical applications throughout
this analysis. In particular, we discuss peculiarities of their branching
properties. Finally, closed-form expressions relating all Casimir operators of
two groups in a pair are established.
