摘要： Linear wave equations on Hamiltonian lattices with translational invariance are characterized by an eigenvalue band structure in reciprocal space. Flat band lattices have at least one of the bands completely dispersionless. Such bands are coined ?at bands. Flat bands occur in ?ne-tuned networks, and can be protected by (e.g. chiral) symmetries. Recently a number of such systems were realized in structured optical systems, exciton-polariton condensates, and ultracold atomic gases. Flat band networks support compact localized modes. Local defects couple these compact modes to dispersive states and generate Fano resonances in the wave propagation. Disorder (i.e. a ?nite density of defects) leads to a dense set of Fano defects, and to novel scaling laws in the localization length of disordered dispersive states. Nonlinearities can preserve the compactness of ?at band modes, along with renormalizing (tuning) their frequencies. These strictly compact nonlinear excitations induce tunable Fano resonances in the wave propagation of a nonlinear ?at band lattice.