Non-commutative conditional expectations and martingales arise inthe setting of von Neumann algebras, which are the naturalframework for non-commutative measure theory and integration.Analogues of classical martingale inequalities such asBurkholder-Gundy's square function inequalities and Doob'sinequality have recently been established for martingales innon-commutative $L_p$-spaces by Junge, Pisier and Xu. They alsoproved the analogue of the classical duality between $H^1$ and$BMO$ of martingales. We will discuss interpolation properties ofnon-commutative $BMO$ and show that it is a natural substitute for$L_infty,.$ As an application we establish boundedness ofnon-commutative martingale transforms.