This book presents a pedagogical introduction (i.e., a text written at a relatively simple level, which makes it accessible to students) to the topic of topological solitons, in the form of kinks and domain walls, in various classical and quantum theories. The book starts with the one-dimensional kink solutions to the simplest model of the so-called \(\varphi^4\) type, and proceeds to more complex models with intrinsic symmetries (multicomponent systems, in particular ones featuring the SU(5) symmetry), \(3+1\)-dimensional domain walls, and kinks in discrete models (solitons in the Toda lattice). Dynamical properties of the kink solutions and solutions close to them are considered in detail, including the formation and stability problems, zero modes and condensation of perturbations on kinks, and others.
A special chapter is dealing with kink states in quantum field theories, which includes the consideration of quantized excitations, the equivalence between topological solitons in the sine-Gordon and massive Thirring models in one dimension (both models are integrable, in their classical and quantum forms alike), quantum kinks on lattices, etc. Another chapter is dealing with domain walls in the gravity theory and cosmology. The last chapter offers an overview of kink-like states and excitations observed in the experiment, including polyacetylene, long Josephson junctions, and some others.