Elemental materials tend to be metallic under pressure and many of them are expected to be also superconductors. Up to now, among 92 elements in the periodic table, there are 30 known elemental superconductors at ambient pressure and 22 more under high pressure. Hydrogen-dense materials recently started to receive increasing interest from the scientific community with a particularly attractive feature of them being high-temperature superconductivity under pressure. Here, we propose to predict novel high-pressure phases of materials such as elements and hydrogen-dense materials in a systematic manner through ab initio random structure search. For rare earths elements, we will use the predicted structures to calculate electron-phonon couplings and the critical temperature of superconductivity. Under high pressure, it is not unusual to find phases which energetically almost degenerated and which are nearly indistinguishable within the accuracy of density functional theory calculations. In these difficult cases, we will resort to the application of quantum Monte Carlo calculations to resolve the enthalpy difference between competing crystal structures. We expect to find new phases of elements and hydrogen-dense materials under pressure, possibly possessing high Tc under pressure. Based on quantitative matching between theory and experiments (where experimental data is available), we can complete the picture of a universal behavior of Tc under pressure for materials including rare earths elements. These results will be greatly beneficial to a broad range of research areas as a reference and for this reason, this project will be of enormous value for Sweden's scientific competitiveness, especially in the increasingly important area of computational condensed matter theory research.