Many integrable field theories are known to have topological excitations, so-called solitons in their spectra. Although their existence is not directly tied to integrability, the existence of infinitely many local conserved quantities plays an important role in their dynamics. Recently we investigated whether such excitations can survive a nonintegrable perturbation. We found a very simple and comprehensive picture for models that are formulated as perturbations of conformal field theories. It turns out that in many cases solitons are confined by the nonintegrable perturbations, but there are large classes of models for which nontrivial topological excitations do survive.