Maximal inequalities are of paramount importance in analysis. Here ``analysis" is understood in a wide sense and includes functional/harmonic analysis, ergodic theory and probability theory. Consider, for instance, the three fundamental examples: Hardy-Littlewood maximal function; Maximal ergodic function; Maximal martingale function. All three maximal functions satisfy the classical inequality, which is due to Hardy-Littlewood, Dunford-Schwartz and Doob.

We will consider in this survey talk the analogues of all these classical inequalities  in noncommutative analysis. Then the usual $L_p$-spaces are replaced by noncommutative $L_p$-spaces. The theory of noncommutative martingale/ergodic inequalities was remarkable developed in the last 20 years. Many classical results were successfully transferred to the noncommutative setting. This theory has fruitful interactions with operator spaces, quantum probablity  and noncommutative harmonic analysis. We will discuss some of these noncommutative results and explain certain substantial difficulties in proving them.

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