Solitons are particle-like solutions to dispersive evolution equations whose shapes persist as time evolves. In some situations, these solitons appear due to the balance between nonlinear effects and dispersion, in other situations their existence is related to topological properties of the model. Broadly speaking, they form the building blocks for the long-time dynamics of dispersive equations. In this talk I will present forthcoming joint work with J. Luehrmann (TAMU) on long-time decay estimates for perturbations of the soliton for the 1D focusing cubic Klein-Gordon equation (up to exponential time scales), and I will discuss our previous work on the asymptotic stability of the sine-Gordon kink under odd perturbations. While these two problems are quite similar at first sight, we will see that they differ by a subtle cancellation property, which has significant consequences for the long-time dynamics of the perturbations of the respective solitons.