Non-Hermitian topology represents a significant departure from conventional Hermitian systems by incorporating gain, loss, or non-reciprocal couplings. This framework has led to the discovery of novel phenomena such as exceptional points, non-Hermitian skin effects, and unconventional bulk-boundary correspondences. Unlike Hermitian topological materials, where energy conservation and unitary evolution dominate, non-Hermitian systems exhibit complex eigenvalue spectra and unique dynamical behaviors. These features have been experimentally realized in photonic, acoustic, and electronic platforms, offering new avenues for device applications and fundamental research.

Exceptional points are a hallmark of non-Hermitian systems, occurring when eigenvalues and their corresponding eigenvectors coalesce. These spectral singularities arise in parameter spaces where gain and loss are carefully balanced or non-reciprocal interactions are introduced. Mathematically, an exceptional point is a branch point singularity in the complex eigenvalue plane, leading to a square-root topology in the vicinity of the singularity. The presence of exceptional points can drastically alter system dynamics, enabling enhanced sensitivity to perturbations and chiral mode switching. Experimental realizations have been achieved in coupled laser systems, microwave cavities, and optomechanical setups, where the encircling of exceptional points results in non-adiabatic transitions and asymmetric state transfer.

Gain and loss play a central role in non-Hermitian topology, breaking the conventional symmetry constraints of Hermitian systems. Parity-time symmetric systems, where gain and loss are spatially balanced, represent a well-studied class of non-Hermitian models. In such systems, the spectrum can transition from purely real to complex eigenvalues as the gain-loss parameter exceeds a critical threshold, leading to spontaneous symmetry breaking. This transition has been observed in photonic waveguides, where judiciously placed optical amplifiers and attenuators create synthetic gain-loss landscapes. Beyond parity-time symmetry, arbitrary gain-loss distributions can give rise to higher-order exceptional points, anomalous edge states, and non-Hermitian band braiding. These effects are not merely perturbations of Hermitian physics but represent entirely new topological phases.

The non-Hermitian skin effect is another striking phenomenon, where bulk states localize at boundaries due to non-reciprocal couplings or asymmetric hoppings. This effect violates the conventional bulk-boundary correspondence, as the topological invariants calculated under periodic boundary conditions fail to predict edge states under open boundaries. The skin effect has been demonstrated in one-dimensional lattices with asymmetric hopping, where all eigenstates accumulate at one end of the system. Higher-dimensional generalizations involve non-reciprocal pumping in two-dimensional lattices, leading to directional amplification and chiral transport. Experimental platforms such as active photonic arrays and electric circuits have confirmed these predictions, showcasing the robustness of the skin effect against disorder.

Non-Hermitian topology also manifests in open quantum systems, where system-environment interactions lead to effective non-Hermitian descriptions. Quantum jumps and dissipation can induce topological phase transitions, with the Liouvillian spectrum replacing the Hamiltonian spectrum as the relevant object of study. Dissipative topological insulators and superconductors exhibit unique features such as dissipative edge modes and non-Hermitian Majorana fermions. These systems bridge the gap between topological matter and non-equilibrium statistical mechanics, offering insights into the stability of topological phases in realistic environments. Cold atom experiments and superconducting qubit arrays have provided preliminary evidence for these phenomena, though challenges remain in achieving full control over dissipative processes.

Experimental realizations of non-Hermitian topology span multiple physical domains. Photonic platforms, including coupled waveguides and ring resonators, allow precise engineering of gain, loss, and non-reciprocity through optical pumping and tailored refractive indices. Acoustic metamaterials utilize active components such as speakers and microphones to create synthetic non-Hermitian potentials, enabling the observation of exceptional points and skin effects. Electronic circuits with active elements like operational amplifiers provide a versatile testbed for non-Hermitian lattice models, with the advantage of tunable parameters and real-time measurements. Recent advances in solid-state systems, such as magneto-optical films and driven quantum dots, suggest that non-Hermitian topology may also emerge in naturally occurring materials under external drives or environmental coupling.

The interplay between non-Hermiticity and nonlinearity introduces additional complexity and richness to topological phenomena. Nonlinear effects can stabilize or destabilize exceptional points, create non-Hermitian solitons, and induce topological transitions through self-tuning of system parameters. Optical Kerr media and Bose-Einstein condensates with controlled dissipation serve as promising platforms for exploring these effects. The combination of non-Hermitian topology and nonlinearity remains an active area of research, with potential applications in robust signal processing and topological lasers.

Device applications of non-Hermitian topology leverage its unique features for improved performance and functionality. Sensors operating near exceptional points exhibit enhanced sensitivity due to the square-root topology of eigenvalue splitting, enabling detection of minute perturbations in refractive index or molecular concentration. Topological lasers exploit the interplay between gain and non-Hermitian band structure to achieve single-mode lasing and directional emission. Non-reciprocal devices based on the skin effect provide robust isolation and filtering without relying on magnetic materials, advantageous for integrated photonic circuits. The design principles of these devices differ fundamentally from their Hermitian counterparts, requiring new approaches to optimization and fabrication.

Theoretical advances continue to expand the scope of non-Hermitian topology, with recent work exploring non-Abelian exceptional points, non-Hermitian higher-order topology, and Floquet non-Hermitian systems. Classification schemes based on generalized symmetries and Clifford algebras have been developed to encompass the diverse manifestations of non-Hermitian topology. Numerical techniques tailored for non-Hermitian systems, such as non-unitary time evolution and complex band structure calculations, have become essential tools for predicting and analyzing these phenomena.

Challenges remain in fully characterizing and controlling non-Hermitian topological systems, particularly in the presence of disorder and interactions. The stability of non-Hermitian phases under realistic conditions, the role of quantum fluctuations, and the scalability of non-Hermitian devices are open questions requiring further investigation. As experimental techniques advance, the exploration of non-Hermitian topology is expected to yield deeper insights into non-equilibrium physics and inspire novel technological applications. The field stands at the intersection of fundamental mathematics and applied physics, with its potential only beginning to be realized.