ABHYANKAR 82nd BIRTHDAY CONFERENCE SPEAKER: Natalia Korskova (St. Petersburg) TITLE: On finite morphisms and curve singularities ABSTRACT: The talk contains a survey of different achievements and approaches to ramification theory for extensions, produced by the finite morphism of schemes in positive characteristic. Study of singularities of arcs is given, including semi-continuity of ramification invariants. Open problems in more general case are discussed. Challenging theory of model morphisms is introduced.