In various areas of applied mechanics, there are instances where it is necessary or beneficial to represent the behavior of a mechanical system on an artificial boundary, or artificial interface, which is introduced into the system. Examples include, among others, the computational treatment of mechanical problems in infinite media, the solution of crack problems in fracture mechanics, the dynamical analysis of a mechanical system attached to a number of smaller subsystems, iterative domain decomposition methods, and the mathematical formulation of inverse problems. The representation of the solution on the interface may be approximate or exact. This article is concerned with exact representations. It explains the benefit in using such representations, compares them to approximate representations in various respects, surveys work that has been done in this field, and highlights applications in applied mechanics. It is the author’s opinion that despite the fact that approximate interface representations are more popular than exact ones, the latter have definite advantages in many situations. References cited in this review article number 163.