Berkovich spaces are analogues of complex manifolds when the complex numbers are replaced by a non-Archimedean field, that is, a field satifying the strong triangle inequality.

I will discuss two instances where Berkovich spaces naturally appear within complex geometry. The first concerns the Yau--Tian--Donaldson conjecture, on the existence of Kähler--Einstein metrics on Fano manifolds. The second situation appears in the context of degenerations of Calabi--Yau manifolds, and features conjectures by Strominger--Yau--Zaslow, and Kontsevich--Soibelman.

This is based on joint work with R. Berman, S. Boucksom, J, Hultgren, E. Mazzon, and N. McCleerey.