In a homotopy setting, i.e. of fibrations = maps p:E--> B with the homotopy lifting property, parallel transport and holonomy can be defined without a connection and in terms of morphisms from the space of paths or based loops without passing to homotopy. Closely related is the notion of (strong or $\infty$) homotopy action, which has variants under a variety of names. My aim is to impose some order on this zoo of concepts and names with major emphasis on the examples coming from fibrations

Inspired by recent extensions in the smooth setting of parallel transport to representations of $Sing_{smooth}(B)$ on a smooth fibre bundle, I revisit the development of a notion of `parallel´ transport in the topological setting of fibrations with the homotopy lifting property and then extend it to representations of $Sing(B)$ on such fibrations.