Beta ensembles are a family of random matrix models parameterized by a real value β >0. Two recent results that offer new structural insights into beta ensembles will be presented. The first part of the talk focuses on stochastic domination in Hermite and Laguerre beta ensembles,where we establish finite-n comparisons of the largest eigenvalues as β varies. These results strengthen and generalize known asymptotic stochastic orderings of Tracy-Widom distributions. In the second part, we present log-concavity phenomena in several discrete and continuous Coulomb gas models. This part of the talk is based on joint work with Manjunath Krishnapur and Mokshay Madiman.