Random matrix theory, in the most general sense, is the study of matrices with random entries, with some constraints on these entries. Carefully choosing ‘nice’constraints leads to many types of random matrix theories, with some key cases having a rich theory that is deeply entrenched in the literature. In this thesis, we review the Gaussian β ensembles before moving onto the Laguerre β ensembles. The symbol β denotes a positive real number indexing the ensembles, with the further significance that the special cases β= 1, 2, and 4 correspond to the orthogonal, unitary, and symplectic symmetry respectively. Our review focuses on the cases β= 1, 2, and 4, with brief mention of the general case. In our treatment of the Laguerre β ensembles, we use the loop equation formalism to investigate the resolvents, which relate to the moments of the eigenvalue densities. We then investigate the special case β= 2 through differential equations for the densities and resolvents. The discussion draws on the literature where similar techniques have been applied to the Gaussian β ensembles.