In the present paper we define Hardy spaces with variable exponents on Rn by the grand maximal function, and then investigate their several properties. The present paper will connect harmonic analysis with function spaces with variable exponents. We 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. The present paper covers local Hardy spaces with variable exponents. © 2012 Elsevier Inc.