Scattering resonances have significant applications across various fields of science and engineering. The associated problem is nonlinear and defined on an unbounded domain. In this talk, we discuss recent advancements in the computation of scattering resonances (poles of the scattering operator) for compact obstacles. We begin by introducing the highly accurate Nyström method for the boundary integral formulation, followed by a finite element method combined with Dirichlet-to-Neumann mapping. The convergence is proved using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. The nonlinear matrix eigenvalue problem is solved using a parallel multistep spectral indicator method, with numerical examples serving as benchmarks. The talk concludes with new results on inverse problems related to scattering resonances.

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