Lecture notes on the circular law

Abridged and updated version of the Probability Survey MR2908617, prepared at the occasion of the AMS short course on Random Matrices, organized by Van H. Vu for the 2013 AMS-MAA Joint Mathematics Meeting held in January 9-13 in San Diego, CA, USA. Other contributions by Alice Guionnet, Alan Edelman & Brian Sutton & Yuyang Wang, Mark Rudelson, Terence Tao & Van Vu.The circular law theorem states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. In these expository notes, we present a proof in a Gaussian case, due to Mehta and Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. We also discuss some related models and open problems