In this talk we will outline recent developments in descritptive inner model theory that have applications in the theory of forcing axioms, in determinacy theory and in infintary combinatorics. In particular, we will concentrate on the following four themes. 1. Cofinality of Theta^{L(uB)} in models of Forcing Axioms and in models of Sealing. We will show that Sealing doesn't decide thie value of this cofinality. In particular, letting kappa be this cofinality, we will show that T_i=Sealing+kappa=omega_i is consistent for i=1, 2, and 3. This is joint work with Douglas Blue and Matteo Viale. 2. We will show that omega_1 is <Theta-Berkeley in models of AD^+. This is a joint work with Douglas Blue. 3. For each n< omega, we will force the theory ZFC+MM^{++}(c)+for all i\leq n+2( failure of square_{omega_i} + failure of square(omega_i)) over a model of determinacy. This is a joint work with Paul Larson and Douglas Blue. 4. We will outline some basic theory of Nairian Models that suggest the possibility of forcing even stronger ideals on omega_1. Clause 3 above has a consequence that the K^c constructions with 2^omega closed background certificates can consistently fail to converge.