  We give a survey of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu on
the connections between the geometry of Berkovich spaces over the field of
Laurent series and the birational geometry of one-parameter degenerations of
smooth projective varieties. The central objects in our theory are the weight
function and the essential skeleton of the degeneration. We tried to keep the
text self-contained, so that it can serve as an introduction to Berkovich
geometry for birational geometers.
