Consider the eigenvalues of a large, square non-normal matrix. For example, consider the matrix which is zero everywhere, except for ones on the entries directly above the diagonal. The eigenvalues of this matrix are all zero. However, due to floating point errors, certain numerical software packages compute the eigenvalues to be nearly equidistributed on the unit circle in the complex plane. In this talk, I will explain how one can model these errors using small, random perturbations, and I will give a probabilistic explanation for why the eigenvalues appear near the unit circle. More generally, I will discuss several theoretical results which describe the behavior of the eigenvalues of randomly perturbed non-normal matrices. This talk is based on joint work with Phillip Matchett Wood.