Abstract: Scattering theory studies the large time behavior of quantum-mechanical systems. Besides the dynamics, it has important spectral implications, say, in particular, the stability of absolutely continuous spectrum. In this talk, I will first give an introduction to the spectral theory and scattering theory of self-adjoint operators. Then, I will present some classical results about Schrodinger operators with decaying potentials as well as some open problems about Schrodinger operators with long-range potentials. Classical and recent results about Schrodinger operators with sparse potentials will also be presented.