This text is a concise introduction to the partial differential equations which change from elliptic to hyperbolic type across a smooth hypersurface of their domain. These are becoming increasingly important in diverse sub-fields of both applied mathematics and engineering, for example:
The heating of fusion plasmas by electromagnetic waves
The behaviour of light near a caustic
Extremal surfaces in the space of special relativity
The formation of rapids; transonic and multiphase fluid flow
The dynamics of certain models for elastic structures
The shape of...