Following the 2004 Cambridge book by Manton and Sutcliffe Topological Solitons, we recall three classical field theories via their Lagrangians: i) scalar fields on the line associated with kinks (the topological invariant is the charge) ii) wave maps in the energy critical regime associated with harmonic maps (the invariant is the degree) iii) Ginzburg-Landau (non-magnetic or non-gauged) and abelian Yang-Mills-Higgs models (magnetic or gauged) associated with vortices (the invariant is the degree). The gradient flow associated with the underlying Hamiltonian leads to dissipative dynamics in the form of a heat equation, while the most common conservative time evolutions are the Schrödinger flow (with Galilei symmetry), respectively the wave or Klein-Gordon flow (with Lorentz symmetry). In each of these infinite-dynamical systems we would like to describe or even classify the long-term behavior of trajectories. This is a deeply challenging problem as a multitude of phenomena might arise (breathers and wobbling kinks in the sine-Gordon equation, multi-kink/antikink solutions in the phi-4 model, bubbling in the harmonic map heat flow, vortex splitting and vortex collisions in Ginzburg-Landau). As stationary solutions, solitons and their moduli space play a fundamental role in the complicated dynamics. A starting point here is the question of asymptotic stability of these equilibria. While the past 20 years have seen dramatic advances, we are still far from a complete understanding. We will survey some of the work in this direction - both past and ongoing - which combines techniques from elliptic PDEs, the spectral and scattering theory of linear differential equations, harmonic analysis and linear dispersive estimates, and nonlinear dispersive equations (space-time resonances, normal forms, vector fields, Fermi Golden Rule).