The universality phenomenon has been observed in many contexts of physics and mathematics. In random matrix theory, the Wigner-Dyson-Mehta universality conjecture asserts that the local eigenvalue statistics of large random matrices are independent of the details of the distribution of the matrix ensemble and only depend on a few universal parameters such as the symmetry class of the matrix.  In this talk, I will give an overview of recent developments on the universality conjecture for random matrix ensembles and related particle systems.