Fano resonances in nanoscale structures are a phenomenon where interference between discrete and continuous states leads to asymmetric line shapes in scattering processes. This concept, originally introduced by Ugo Fano in 1935, has been widely applied in various physical systems, including light-matter interactions, charge transport, and quantum systems. The Fano resonance is characterized by an asymmetric profile due to the interaction between a discrete state and a continuum of propagation modes. The asymmetry parameter, q, determines the shape of the resonance, with q = 0 corresponding to a symmetric profile and larger values leading to asymmetric shapes. The Fano resonance has been observed in numerous systems, such as light scattering by nanoparticles, plasmon scattering in Josephson junctions, and charge transport through quantum dots. Theoretical models, including the Fano-Anderson model, have been developed to explain these phenomena, showing how interference between different paths leads to resonant enhancement or suppression of transmission. Nonlinear Fano resonances have also been studied, where the intensity of the input wave affects the resonance position and width. In complex geometries, Fano resonances can arise from multiple scattering paths, leading to asymmetric transmission profiles. The presence of nonlinearities can further modify the resonance characteristics, allowing for tunable Fano resonances by adjusting the input intensity. Additionally, Fano resonances have been observed in systems with complex dynamics, such as those involving discrete breathers and solitons, where the interaction between different wave modes leads to interference effects. The study of Fano resonances in nanoscale structures highlights the importance of interference phenomena in various physical systems, providing insights into the underlying mechanisms of resonance and scattering processes. The application of Fano resonances in optical devices, quantum dots, and other nanoscale systems demonstrates their significance in both fundamental physics and technological applications.Fano resonances in nanoscale structures are a phenomenon where interference between discrete and continuous states leads to asymmetric line shapes in scattering processes. This concept, originally introduced by Ugo Fano in 1935, has been widely applied in various physical systems, including light-matter interactions, charge transport, and quantum systems. The Fano resonance is characterized by an asymmetric profile due to the interaction between a discrete state and a continuum of propagation modes. The asymmetry parameter, q, determines the shape of the resonance, with q = 0 corresponding to a symmetric profile and larger values leading to asymmetric shapes. The Fano resonance has been observed in numerous systems, such as light scattering by nanoparticles, plasmon scattering in Josephson junctions, and charge transport through quantum dots. Theoretical models, including the Fano-Anderson model, have been developed to explain these phenomena, showing how interference between different paths leads to resonant enhancement or suppression of transmission. Nonlinear Fano resonances have also been studied, where the intensity of the input wave affects the resonance position and width. In complex geometries, Fano resonances can arise from multiple scattering paths, leading to asymmetric transmission profiles. The presence of nonlinearities can further modify the resonance characteristics, allowing for tunable Fano resonances by adjusting the input intensity. Additionally, Fano resonances have been observed in systems with complex dynamics, such as those involving discrete breathers and solitons, where the interaction between different wave modes leads to interference effects. The study of Fano resonances in nanoscale structures highlights the importance of interference phenomena in various physical systems, providing insights into the underlying mechanisms of resonance and scattering processes. The application of Fano resonances in optical devices, quantum dots, and other nanoscale systems demonstrates their significance in both fundamental physics and technological applications.