In the paper, we establish and study Hardy spaces with variable exponents on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss, where d may have no any regularity property and µ fulfills the doubling property only. First we introduce the Hardy spaces with variable exponents H p(·) (X) by using the wavelet Littlewood–Paley square functions and give their equivalent characterizations. Then we establish the atomic characterization theory for H p(·) (X) via the new Calderón-type reproducing identity and the Littlewood–Paley–Stein theory. Finally, we give a unified method for defining these variable Hardy spaces H p(·) (X) in terms of the same spaces of test functions and distributions. More precisely, we introduce the variable Carleson measure spaces CMO p(·) L 2 (X) and characterize the variable Hardy spaces via the distributions of CMO p(·) L 2 (X).