  Possible short and semi-short representations for $\N=2$ and $\N=4$
superconformal symmetry in four dimensions are discussed. For $\N=4$ the well
known short supermultiplets whose lowest dimension conformal primary operators
correspond to $\half$-BPS or ${1\over 4}$-BPS states and are scalar fields
belonging to the $SU(4)_r$ symmetry representations $[0,p,0]$ and $[q,p,q]$ and
having scale dimension $\Delta =p$ and $\Delta = 2q+p$ respectively are
recovered. The representation content of semi-short multiplets, which arise at
the unitarity threshold for long multiplets, is discussed. It is shown how, at
the unitarity threshold, a long multiplet can be decomposed into four
semi-short multiplets. If the conformal primary state is spinless one of these
becomes a short multiplet. For $\N=4$ a ${1\over 4}$-BPS multiplet need not
have a protected dimension unless the primary state belongs to a $[1,p,1]$
representation.
