  We study reproducing kernel Hilbert and Pontryagin spaces of slice
hyperholomorphic functions which are analogs of the Hilbert spaces of analytic
functions introduced by de Branges and Rovnyak. In the first part of the paper
we focus on the case of Hilbert spaces, and introduce in particular a version
of the Hardy space. Then we define Blaschke factors and Blaschke products and
we consider an interpolation problem. In the second part of the paper we turn
to the case of Pontryagin spaces. We first prove some results from the theory
of Pontryagin spaces in the quaternionic setting and, in particular, a theorem
of Shmulyan on densely defined contractive linear relations. We then study
realizations of generalized Schur functions and of generalized Carath'eodory
functions.
