  We prove the analogue of the classical Burkholder-Gundy inequalites for
non-commutative martingales. As applications we give a characterization for an
Ito-Clifford integral to be an $L^p$-martingale via its integrand, and then
extend the Ito-Clifford integral theory in $L^2$, developed by Barnett,
Streater and Wilde, to $L^p$ for all $1<p<\infty$. We include an appendix on
the non-commutative analogue of the classical Fefferman duality between $H^1$
and $BMO$.
