Analytic mappings between Riemann surfaces are very natural objects in complex analysis. Corresponding to the classical univalent functions we have the class of injective holomorphic mappings --- i.e., conformal embeddings --- of a Riemann surface into another. We find indeed a number of analogies between them. On the other hand, because of the non-planarity of the domain surface, we face some new problems which we have never encountered in the classical theory. We discuss various problems concerning the conformal embeddings.