Weighted backward shifts are among the most important classes of operators. Their dynamic behaviour has been well understood since the work of Salas. The need for creating more sophisticated examples of operators yet preserving some of the nice properties of weighted backward shifts has motivated the researchers to generalise the notion of a backward shift in several different directions. In the present talk, we will introduce weighted shifts on trees, weighted shifts on graphs, as well as an abstraction of those which we call generalized shifts. We will try to indicate the common properties enjoyed by all those shifts and mention some of their differences. The talk will be based on joint works with Karl Grosse-Erdmann, Quentin Menet, Anton Baranov and Andrei Lishanskii.