This chapter is a survey on the developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. The paper gives an overview on rational functions and their realizations, on the Hardy space of the unit ball and on the half space of quaternions with positive real part, on Schur multipliers, also discussing related results. For purpose of comparison, the paper also presents another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation, most of the article is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions.