Nonlinear random matrices have emerged as an important area of interest in statistics and machine learning. In recent years, the study of nonlinear random matrix theory has proven valuable in providing rigorous explanations for empirically observed phenomena in deep learning. From a random matrix perspective, this creates new challenges in handling nonlinearity and dependence. In this talk, I will survey recent developments in the spectral analysis of nonlinear random matrices and their applications. This includes discussions on random feature models, kernel random matrices, and random geometric graphs.