Aspects of defects in integrable quantum field theory

Defects are ubiquitous in nature, for example, in the form of dislocations, shocks, bores, or impurities of various kinds, and their descriptions are an important part of any physical theory. But the following question can be asked. What types of defect are allowed, and what are their properties if maintaining integrability within an integrable field theory in two-dimensional space-time is required? We consider a collection of ideas and questions connected with this problem, including examples of integrable defects and the curiously special roles played by energy-momentum conservation and Bäcklund transformations, solitons scattering on defects, and some interesting effects in the framework of the sine-Gordon model, defects in integrable quantum field theory, and the construction of transmission matrices. In conclusion, we remark on algebraic considerations and future research directions.