We show that the nearly uniform mode generated by the confluence of Bose-Einstein condensate ͑BEC͒ magnon pairs produced by microwave radiation in a film of yttrium iron garnet ͑YIG͒ is a coherent magnon state. This state corresponds to a macroscopic precessing magnetization which emits a superradiant microwave signal as a result of the cooperative action of the spins. The theory explains quantitatively recent experimental observations of microwave emission from a BEC of magnons in a YIG film when the driving power exceeds a critical value.In a recent series of papers Demokritov et al. 1-3 reported the experimental observation of quasiequilibrium Bose-Einstein condensation ͑BEC͒ of magnons at room temperature in films of yttrium iron garnet ͑YIG͒ driven by microwave radiation. When magnetized in the plane these films have a dispersion relation ͑frequency k versus wave vector k͒ for magnons propagating along the field that exhibits a minimum frequency k0 =2f min at k 0 ϳ 10 5 cm −1 . Magnon pairs with opposite wave vectors pumped parametrically in large numbers by the microwave field are redistributed over the spectrum creating a hot magnon gas that remains decoupled from the lattice for some time due to the relatively slow spin-lattice relaxation. The experiments show that if the microwave power exceeds a certain threshold value the intensity of the light scattered by magnon with frequency k0 increases by orders of magnitude indicating the formation of a Bose-Einstein condensate. [1][2][3] We have shown 4 that the cooperative action in an interacting magnon gas in a YIG film creates a coherent magnon state at k0 when the magnon population exceeds a critical density. The coherent state corresponds to a macroscopic precessing magnetization providing theoretical support for the experimentally observed Bose-Einstein condensate of magnons. Since the wavelength of the associated spin wave is on the order of 10 −5 cm, the direct microwave radiation from the BEC is negligible. However, as observed by Dzyapko et al.,5 if the applied in-plane static field has a value such that the frequency of the k Ϸ 0 magnon is 0 =2 k0 , a microwave radiation is generated by k Ϸ 0 magnons created by pairs of BEC magnons k 0 ,−k 0 through a three-magnon confluent process. The k Ϸ 0 value is necessary for emission because the wave number of electromagnetic radiation with frequency of 1.5 GHz, as in the experiments, 5 is k =2 / f Ϸ 0.3 cm −1 . In this Rapid Communication we show that the k Ϸ 0 magnons created by the BEC are coherent magnons states, which correspond to a nearly uniform magnetization precessing with frequency 0 and generating a microwave signal. The microwave emitted by the collective action of the spins is identified with superradiance. We show that this process occurs when the driving microwave power exceeds a critical value and that the theoretical results for the emitted signal power agree very well with the experimental data. 5 The term superradiance was introduced by Dicke 6 in 1954 to designate the type of spontaneous ...