In this talk, I will consider the problem of computing resonances in open systems. I will first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall see that these both of these steps lead to eigenvalue convergence to the desired resonance values and are free from spirious computational eigenvalues provided that the size of computational domain is sufficiently large and the mesh size is sufficiently small. We illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.