In this talk we will discuss the idea of flowing maps to minimal surfaces by evolving them with a suitably defined gradient flow of the Dirichlet energy. In the first part of the talk we will recall some general features of the classical Plateau problem of finding (parametrized) minimal surfaces in Euclidean space with prescribed boundary as well as discuss how to flow maps that are parametrised over the cylinder. In the second part of the talk we will then discuss the more general problem of flowing to minimal surfaces of more general (topological) type in Riemannian manifolds.