Syntheses, heat treatments, structures and superconducting properties of Hg-based superconducting Cu mixed oxides with the general formula HgBa2Can-1CunO2n+2+δ are presented and discussed. The increase of the superconducting transition temperature with n occurs up to the third member. The heat-treatment induces distinct changes of Tc for different members of the series. Superconductivity in Hg-1212 and Hg-1201 exists in underdoped and overdoped regions. In the structures of Hg-based superconductors the (CuO2) layers have an undistorted ideal arrangement, which could be responsible for the existence of superconductivity at temperatures around 120-135 K.