In analogy to Simpson's non-abelian Hodge theory over the complex numbers, the p-adic Simpson correspondence over non-archimedean fields like C_p aims to relate p-adic representations of the étale fundamental group of a smooth proper rigid space X to Higgs bundles on X. In this talk, I will introduce p-adic moduli spaces for either side of the correspondence, and explain how these can be compared by way of a non-abelian generalisation of the Hodge-Tate sequence. This allows one to construct new geometric incarnations of the p-adic Simpson correspondence, and to interpret the choices necessary for its formulation in a geometric fashion.