I will give an overview on results about the higher ramification theory of finite Galois extensions, mainly, but not only, those of prime degree, for arbitrary valuations. I willtalk about ramification groups, ramification ideals, ramification jumps, norms and traces of these ideals, Kähler differentials and their annihilators, and Dedekind differents. Several of these objects are used for the classification of defects of Galois defect extensions of prime degree. This in turn is used to prove results about the class of deeply ramified fields, which also contains the perfectoid fields.