In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices. In these two talks, this will serve as our starting point as we surveyed the historical developments in this field. We will end with a discussion of the proof of the local semicircle law of Erdos, Schlein and Yau and the four moment theorem by Tao and Vu. We will also discuss some of the open problems in the study of random matrices if time permits.