The main objective of the proposal is to find new ways to increase substantially the critical temperature of a superconductor, determine the experimental feasibility of the theoretical predictions and, in collaboration with experimentalists, find the experimental setting with an optimal enhancement of superconductivity. In order to pursue this goal we investigate: (1) finite size effects in different type of superconductors - Pb, fullerides, cuprates and iron pnictides -, (2) novel forms of superconductivity based on Efimov states, (3) superconductivity in systems that do not thermalize, namely, with quasiparticle distributions different from Fermi-Dirac. In (1) the main task is to identify the grain size and shape, amount of disorder and type of interaction for which Tc increases the most and assess its practical/experimental feasibility and its potential for technological applications. In (2) and (3) we first aim to identify the properties of the superconducting state and its robustness under thermal and quantum fluctuations and then explore the optimal setting to observe an enhancement of superconductivity. Moreover, in order to gain theoretical understanding of some of these strongly interacting systems, we employ the AdS-CFT correspondence (see background) in order to put forward a quantitative, though effective, description of certain aspects of high temperature superconductors and systems close to quantum critical points.