In recent years, machine learning has motivated the study of what one might call "nonlinear random matrices." This broad term includes various random matrices whose construction involves the *entrywise* application of some deterministic nonlinear function, such as ReLU. We study one such model, an entrywise nonlinear function of a sample covariance matrix f(X*X), typically called a "random inner-product kernel matrix" in the literature. A priori, entrywise modifications of a matrix can affect the eigenvalues in complicated ways, but recent work of Lu and Yau established that the large-scale spectrum of such matrices actually behaves in a simple way, described by free probability, when the randomness in X is either uniform on the sphere or Gaussian. We show that this description is universal, holding for much more general randomness in X. Joint work with Sofiia Dubova, Yue M. Lu, and Horng-Tzer Yau.