Self-localized nonlinear excitations (solitons, polarons, and bipolarons) are fundamental and inherent features of quasi-one-dimensional conducting polymers. Their signatures are evident in many aspects of the physical and chemical properties of this growing class of novel materials. As a result, these polymers represent an opportunity for exploring the novel phenomena associated with topological solitons and their linear confinement which results from weakly lifting the ground-state degeneracy. The authors review the theoretical models that have been developed to describe the physics of polyacetylene and related conducting polymers and summarize the relevant experimental results obtained for these materials. An attempt is made to assess the validity of the soliton model of polyacetylene and its generalization to related systems in which the ground-state degeneracy has been lifted.