Random matrix theory is concerned primarily with the eigenvalues and eigenvectors of certain ensembles. The method of moments is a common technique in this area: it goes back to Wigner in 1955, and consists of finding the asymptotic behavior of traces of large powers of matrices. The most common uses are operator norm bounds with high probability, a classical result of Soshnikov from 1998 being edge eigenspectrum universality. The purpose of this talk is highlighting the versatility of this technique by presenting three (more) recent applications, two yielding central limit theorems for eigenvalues, and one perhaps even surprising, related to eigenvectors. No background in random matrix theory is assumed.