In Physics, one of the major goals is to construct good theoretical models that explain the observations in the physical world. Can the spectra of matrices with random entries serve as useful models for some physical systems? Turns out the answer is yes! The spectrum of random matrices serves as useful models for atomic nuclei, classically chaotic quantum systems, and even the zeros of the Riemann Zeta function, among several other applications. In my talk, I will motivate the study of the spectrum of random matrices from the above examples and then cover some basic results in random matrix theory. If time permits, I will discuss some of the recent research directions in the study of random matrices.