It is an old idea of Serre that the classical Jacquet-Langlands correspondence between modular forms and quaternion modular forms can be realised geometrically. In this talk I will discuss an extension of these ideas to Siegel modular forms of genus two and paramodular level. We use this to prove the weight-monodromy conjecture for the Siegel threefold of paramodular level. Moreover we construct a geometric Jacquet-Langlands correspondence between GSp4 and a `definite' inner form, proving a conjecture of Ibukiyama.