Deninger and Werner developed an analogue for $p$-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a $p$-adic curve whose reduction is strongly semi-stable of degree 0 a $p$-adic representation of the \'etale fundamental group of the curve. They asked several questions: whether their functor is fully faithful and what is its essential image; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate filtration; and whether their construction is compatible with the $p$-adic Simpson correspondence developed by Faltings. We will answer these questions in this talk.