Arakelov geometry over adelic curves is a theory of arithmetic geometry over a base field equipped with a family of absolute values parametrized by a measure space. It is a very general setting since any countable field admits natural adelic curve structures fibered over the classic adelic structure of its prime field. It turns out that many classic results can be generalized to this setting. In this talk, I would like to report one of such results : the arithmetic Hilbert-Samuel theorem describing the asymptotic behavior of an adelic line bundle on an arithmetic projective variety. Although the statement is similar to the classic form, the proof need new idea of arithmetic geometry over a trivially valued field. This is a work in collaboration with Atsushi Moriwaki.