  We study the unitary representations of the non-compact real forms of the
complex Lie superalgebra sl(n|m). Among them, only the real form su(p,q|m)
(p+q=n) admits nontrivial unitary representations, and all such representations
are of the highest-weight type (or the lowest-weight type). We extend the
standard oscillator construction of the unitary representations of non-compact
Lie superalgebras over standard Fock spaces to generalised Fock spaces which
allows us to define the action of oscillator determinants raised to non-integer
powers. We prove that the proposed construction yields all the unitary
representations including those with continuous labels. The unitary
representations can be diagrammatically represented by non-compact Young
diagrams. We apply our general results to the physically important case of
four-dimensional conformal superalgebra su(2,2|4) and show how it yields
readily its unitary representations including those corresponding to
supermultiplets of conformal fields with continuous (anomalous) scaling
dimensions.
