A cation exchange method was applied to convert epitaxial TlBa2Ca2Cu3O8 (Tl-1223) and Tl2Ba2Ca2Cu3O10 (Tl-2223) precursor superconducting thin films to epitaxial HgBa2Ca2Cu3O8+δ (Hg-1223) films on LaAlO3 (001) substrates. While the conversion of Tl-1223 to Hg-1223 was partially successful, high-quality epitaxial Hg-1223 films were obtained from Tl-2223 precursor films. A critical transition temperature Tc of up to 132 K has been demonstrated for the Hg-1223 films, which is close to the optimal value of 135 K reported on Hg-1223 bulk samples. The critical current density Jc of the Hg-1223 films was up to 0.25 MA cm−2 at 77 K and self-field. This work represents the first success in achieving the highest-Tc epitaxial Hg-1223 films using the cation exchange method, which are promising candidates for device applications with operational temperatures significantly above the temperature of liquid nitrogen.