By work of Ben-Yaacov complete valued fields with value groups embedded in the real numbers can be viewed as metric structures in continuous logic. For technical reasons one has to consider the projective line ov er such a field rather than the field itself. In this talk we introduce the above setting and give a classification of the elementary classes of metric valued fields in equicharacteristic 0 in terms of their residue field and value group. This can also be se en as an approximate AKE principle. As a second result we give a negative answer to a question of Ben-Yaac ov on the existence of a model companion for metric valued fields enriched with an isometric isomorphism.