The subject of my talk is random symmetric Toeplitz matrices of size n. Assuming that the distinct entries are independent centered random variables with finite $$\gamma$$-th moment ($$\gamma>2$$), I will establish a law of large numbers for the largest eigenvalue. Following the approach of Sen and Virág (2013), in the large $$n$$ limit, the largest eigenvalue rescaled by $$\sqrt{2n\, log(n)}$$ is shown to converge to the limit $$0.8288...$$ . I will explain the background theory and also present some related results. Finally, a numerical analysis is used to illustrate the rate of convergence and the oscillatory nature of the eigenvectors of random Toeplitz matrices.