The Calabi-Yau problem in archimedean and non-archimedean geometry

We describe the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. On complex manifolds the Calabi-Yau problem asks locally for solutions of a PDE of Monge-Ampère type. The complex Calabi-Yau problem was posed by Calabi in 1954 and solved by Yau in 1978. After a presentation of the complex case we give a brief introduction to non-archimedean geometry and report on the recent progress in the non-archimedean case.