Exceptional topology of non-Hermitian systems

Quantum systems that are coupled to an external bath can often be described in terms of a non-Hermitian effective Hamiltonian. In isolated systems with Hermitian Hamiltonians, topological aspects of the band structure, and resulting topological phases, have been of interest. The combination of the two concepts, i.e., topological properties of open systems, leads to qualitatively new effects. This review provides an introduction to these quantum mechanical concepts and their classical analogs, and discusses a number of applications ranging from mechanical metamaterials to dissipative cold-atom systems.