According to Siegmund (1976) two Markov processes $X,Y$ on $\textbb{R}_+$ are dual, if for all $t,x,y\geq 0$
$$\mathbb{P}^x(X_t\leq y) = \PP^y(Y_t\geq x).$$
This duality is a helpful tool in applied probability as it allows (under suitable regularity conditions) to express the stationary law of one of the processes via hitting probabilities of the other process.

We recall a few well-known examples of pairs of dual Markov processes and their applications, add new case-studies, and discuss how to find a dual process in the general context of Lévy-type processes. Further, we will shed some light on the connection between the above duality and the related concept of time-reversal as used in the theory of semimartingales.