The Riemann-Zariski space was introduced by Zariski in the 40's in his work on resolution of singularities. The last fifteen years, it has become clear that Riemann-Zariski spaces are closely related to rigid geometry and Berkovich spaces. There are also more mundane applications of Riemann-Zariski spaces, e.g., they can be used to give a simple proof of Raynaud-Gruson's flatification theorem, the tame etalification theorem (in my WAGS-talk), various compactification theorems and certain weak resolutions of
singularities in characteristic p. I will give an elementary introduction to valuations and Riemann-Zariski spaces and discuss
some of the applications mentioned above.