The direct and inverse scattering problems for the Benjamin-Ono equation were formulated by Fokas and Ablowitz in 1983. Rigorous studies of these problems involving general potentials were done more recently. Wu proved certain important spectral properties of the scattering operators, together with existence and uniqueness of Jost solutions. In this talk, we review the above results, and prove asymptotic formulas of Jost solutions and scattering coefficients as the spectral parameter varies. These formulas are important for understanding the behavior of the inverse scattering problem. The key to the proof is the study of uniqueness of solutions to a certain regularized Fredholm integral equation.