Abstract: The Circular Law states that the empirical eigenvalue distribution of non-Hermitian random matrices with independent entries will converge to the uniform distribution on the complex disk as the size of the matrix tends to infinity. In this talk, I will address the rate of convergence to the Circular Law in terms of a uniform Kolmogorov-like distance. The optimal rate of convergence is determined by matrices with Gaussian entries and is given by \(n^{-1/2}\). Secondly, I will present that also matrices with non-Gaussian entries nearly attain this optimal rate. The method of proof involves a new smoothing inequality for logarithmic polynomials. In a similar fashion, we will also discuss related models such as products of independent matrices and the empirical distribution of the roots of certain random polynomials with independent coefficients. The talk will contain numerous illustrations and numerical simulations supporting some open problems.