One classically addresses Hardy spaces in the context of complex analysis where they are observed on some section of complex plane. In this text, however, a modern treatment of the \(H^p(\mathbb{R}^n)\) space, with \(0<p\leq \infty\), as the class of those tempered distributions whose appropriately defined maximal operator is contained in the \(L^p(\mathbb{R}^n)\) space is laid out. The first major result that is proved claims that the maximal operator used in the definition may be chosen in a couple of equivalent ways. Moreover, we proved that for an arbitrary \(f \in H^p\) a family \(\{a_k\}\) of simpler elements and a sequence of complex coefficients \(\lambda_k\) can be obtained such that \(f=\Sigma_k\lambda_ka_k\) in the \(H^p\) norm; we call the elements \(a_k\) atoms, and the resulting decomposition atomic. Finally, a thorough overview of a particularly interesting space \(H^1\) is given along with the demonstration of the fact that its dual is the space of functions with bounded mean oscillation (BMO).