The p-adic Simpson correspondence aims to give a non-abelian generalisation of the Hodge-Tate decomposition. Following an idea of Faltings, it should relate pro-étale vector bundles on smooth rigid spaces over Cp to Higgs bundles. In this talk, I will first sketch how to construct the p-adic Simpson correspondence for smooth proper rigid spaces. For curves, I will then explain how in joint work with Daxin Xu, we interpret the correspondence more geometrically as a twisted isomorphism between the moduli stacks of either side.