In this talk, I will give a leisurely introduction to the theory of classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory. The first step is to define the geometric theory of absolute values of Q and provide a point-free account of exponentiation (which has already been completed). The next step is to construct the classifying topos of places of Q, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of Spec(Z). This part is still work in progress, but some interesting observations (in particular, regarding whether we ought to view infinity as a point) have already emerged which we would like to share with the community. Joint work with Steve Vickers.