The Hg-based Cu mixed-oxide superconductors belong to the homologous series HgBa2Can-1CunO2n+2+δ. These compounds, besides exhibiting record-high critical temperatures, appear to show favourable behaviour in magnetic fields and their critical current density can be greatly enhanced by neutron irradiation. Another attractive feature of these compounds is their relatively simple structure. This simplicity may lead to a better understanding of the superconducting mechanisms leading to yet higher critical temperatures. The main scientific goals of this project are: the search for new superconductors in the Hg-containing Cu oxide family, especially in view of stabilizing the structural arrangement corresponding to the 164K transition temperature obtained at 35 Gpa; the optimization of ceramic samples; the preparation of single crystals; the determination of the average and local structures by neutron diffraction and ultra high resolution electron microscopy; the measurements of pertinent physical properties, relating both to Tc and Jc, involving magnetic penetration depth, heat capacity, and NMR; the effects of irradiation (neutrons and heavy ions) on Tc and Jc; and the synthesis and chemistry of thin films formed via laser-ablation.