  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 flat bands. Flat bands occur in fine-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 finite 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 flat band modes, along with renormalizing (tuning)
their frequencies. These strictly compact nonlinear excitations induce tunable
Fano resonances in the wave propagation of a nonlinear flat band lattice.
