In this talk, I will present our recent work on the development of computationally efficient propagators and their application to seismic imaging. By incorporating the ideas of efficient absorbing boundary conditions and scattering formalism into a carefully devised double-sweeping scheme, we are able to accurately capture the amplitudes of primary transmission and reflections without solving the two-way wave equation. This approximate solver is applied to the seismic imaging problem in three disparate ways. First, when the algorithm is used in the setting of migration, the ability of capturing steep dips is markedly improved when compared to standard wide-angle one-way propagators. Second, the double-sweeping approximation is shown to be an effective preconditioner for the notoriously difficult Helmholtz solver. Third, when the approximate solver is used in place of the Helmholtz solver for the purposes of computing gradient and Hessian-vector products (in the context of full waveform inversion, FWI), we observe that FWI convergence is almost unaltered, leading to significant reduction in the overall computational cost. This talk would include the details of the double-sweeping formulation, as well as the three applications with appropriate numerical examples. [Joint work with Mehran Eslaminia]