Topological Data Analysis (TDA) is a recent and growing branch of statis- tics devoted to the study of the shape of the data. Motivated by the complexity of the object summarizing the topology of data, we introduce a new topological kernel that allows to extend the TDA toolbox to supervised learning. Exploiting the geodesic structure of the space of Persistence Diagrams, we define a geodesic kernel for Per- sistence Diagrams, we characterize it, and we show with an application that, despite not being positive semi–definite, it can be successfully used in regression tasks.