For a certain class of nonlinear PDEs, called integrable PDEs, it is possible to derive remarkably precise asymptotic formulas for the long-time behavior of the solution by using inverse scattering techniques together with a nonlinear version of the steepest descent method. I will give an introduction to this circle of ideas using the sine-Gordon equation as an illustrative example. At the end, I will present some new results on the topological charge and on the interaction between the asymptotic solitons and the radiation background for the sine-Gordon equation.