  The theory of slice regular functions of a quaternionic variable, introduced
in 2006 by Gentili and Struppa, extends the notion of holomorphic function to
the quaternionic setting. This fast growing theory is already rich of many
results and has interesting applications. In this setting, the present paper is
devoted to introduce and study the quaternionic counterparts of Hardy spaces of
holomorphic functions of one complex variable. The basic properties of the
theory of quaternionic Hardy spaces are investigated, and in particular a
Poisson-type representation formula, the notions of outer function, singular
function and inner function are given. A quaternionic (partial) counterpart of
the classical $H^p$-factorization theorem is proved. This last result assumes a
particularly interesting formulation for a large subclass of slice regular
functions, where it is obtained in terms of an outer function, a singular
function and a quaternionic Blaschke product.
