  Dissipation in mechanics, optics, acoustics, and electronic circuits is
nowadays recognized to be not always detrimental but can be exploited to
achieve non-Hermitian topological phases or properties with functionalities for
potential device applications. As elementary excitations of ordered magnetic
moments that exist in various magnetic materials, magnons are the information
carriers in magnonic devices with low-energy consumption for reprogrammable
logic, non-reciprocal communication, and non-volatile memory functionalities.
Non-Hermitian topological magnonics deals with the engineering of dissipation
and/or gain for non-Hermitian topological phases or properties in magnets that
are not achievable in the conventional Hermitian scenario, with associated
functionalities cross-fertilized with their electronic, acoustic, optic, and
mechanic counterparts, such as giant enhancement of magnonic frequency combs,
magnon amplification, (quantum) sensing of the magnetic field with
unprecedented sensitivity, magnon accumulation, and perfect absorption of
microwaves. In this review article, we address the unified approach in
constructing magnonic non-Hermitian Hamiltonian, introduce the basic
non-Hermitian topological physics, and provide a comprehensive overview of the
recent theoretical and experimental progress towards achieving distinct
non-Hermitian topological phases or properties in magnonic devices, including
exceptional points, exceptional nodal phases, non-Hermitian magnonic SSH model,
and non-Hermitian skin effect. We emphasize the non-Hermitian Hamiltonian
approach based on the Lindbladian or self-energy of the magnonic subsystem but
address the physics beyond it as well, such as the crucial quantum jump effect
in the quantum regime and non-Markovian dynamics. We provide a perspective for
future opportunities and challenges before concluding this article.
