  In this paper we introduce and study some basic properties of the Fock space
(also known as Segal-Bargmann space) in the slice hyperholomorphic setting. We
discuss both the case of slice regular functions over quaternions and also the
case of slice monogenic functions with values in a Clifford algebra. In the
specific setting of quaternions, we also introduce the full Fock space. This
paper can be seen as the beginning of the study of infinite dimensional
analysis in the quaternionic setting.
