Quaternionic de Branges Spaces and characteristic operator function Preliminaries Quaternions and matrices Slice hyperholomorphic functions Rational functions Operator models Structure theorems for H(A;B) spaces J contractive functions The characteristic operator function L() spaces and linear fractional transformations Canonical differential systems. This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case but the noncommutative setting of quaternions shows several interesting new features see e.g. the so called S spectrum and S resolvent operators. In this work we study de Branges spaces namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. The focus of this book is the notion of characteristic operator function of a bounded linear operator A with finite real part and we address several questions like the study of J contractive functions where J is self adjoint and unitary and we also treat the inverse problem namely to characterize which J contractive functions are characteristic operator functions of an operator. In particular we prove the counterpart of Potapovs factorization theorem in this framework. Besides other topics we consider canonical differential equations in the setting of slice hyperholomorphic functions and we define the lossless inverse scattering problem. We also consider the inverse scattering problem associated with canonical differential equations. These equations provide a convenient unifying framework to discuss a number of questions pertaining for example to inverse scattering non linear partial differential equations and are studied in the last section of this book.