Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space H1, which substitute for L1, and the Hardy spaces $H^p$ with $0 < p < 1$, are different in that the latter contains non-regular distributions. Although it will turn out to be an equivalent expression of $L^p$, for $1 < p < \infty$, we can define the Hardy space $H^p$. To have a unified understanding of these situations, we consider and de?ne Hardy spaces with variable exponents on $R^n$. We will connect harmonic analysis with function spaces with variable exponents. We then obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood?Paley characterization. Also, we specify the dual spaces of Hardy spaces with variable exponents. They will turn out to be Campanato spaces with variable growth conditions. We shall allude to local Hardy spaces with variable exponents as well.