Dynamical supersymmetry breaking in a metastable vacuum is an attractive possibility for supersymmetry breaking. In the first part of this book, using Seiberg-Witten techniques, non-supersymmetric metastable vacua in four dimensional N=2 gauge theories softly broken to N=1 by a superpotential term are constructed and studied. Using Thermal Field Theory, it is shown that in SU(2) theory without flavors, the cooling universe can settle in the metastable vacuum under certain conditions. A common feature of UV completions of our current theoretical framework, the Einstein action coupled to the matter content of the Standard Model, are higher derivative corrections to both the gravitational and matter low energy effective actions. In the second part of this book, the local and global thermal phase structure for asymptotically AdS black holes charged under a U(1) group, with both Gauss-Bonnet and quartic field strength corrections, is mapped out for all parameter space in the grand canonical ensemble and at an arbitrary number of dimensions.