The classical (Lebesgue-based) Hardy space Hp is defined as the space of distributions satisfying a certain size condition by demanding the membership of the grand maximal function to the Lebesgue space Lp. Following this template, we introduce the scale of Beurling-Hardy spaces HAp on Ahlfors regular sets by requiring that the grand maximal function belongs to Beurling spaces Ap. In this chapter, we take up the task of developing a functional analytic theory for HAp in which atomic and molecular decompositions play a crucial role (see Section 4.2 and Section 4.3).