In this talk I will introduce the transport equation in the space of k-currents in R^n, that describes the motion of generalized k-dimensional surfaces under a given driving vector field. I will first spend some time explaining the equation, the motivation and the links with well-known classical PDE’s such as the continuity equation and the transport equation. Then I will present some well-posedness results when the driving vector-field has Lipschitz regularity, obtained in collaboration with Paolo Bonicatto (Trento) and Filip Rindler (Warwick). Time permitting I will also discuss some ongoing research concerning ill-posedness, constructing non-unique solutions through convex integration, in collaboration with Francisco Mengual (IMUS Sevilla).