  We report recent advances on noncommutative martingale inequalities
associated with convex functions. These include noncommutative Burkholder-Gundy
inequalities associated with convex functions due to the present authors and
Dirksen and Ricard, noncommutative maximal inequalities associated with convex
functions due to Os\c{e}kowski and the present authors, and noncommutative
Burkholder and Junge-Xu inequalities associated with convex functions due to
Randrianantoanina and Lian Wu. Some open problems for noncommutative
martingales are also included.
