Gamma-ray bursts are emitted by super-powerful ultra-relativistic jets from compact engines, most likely just born black holes or magnetars. A long debated problem is how the jet emits the observed gamma rays. A simplest model resembles the big bang: the opaque hot plasma expands to transparency and releases its thermal photons. The expected spectrum from a radiation-dominated jet is Planckian with a peak around 1 MeV. The observed burst spectra do peak around 1 MeV, however they have non-Plankian shapes, with extended high-energy tails. Physical processes generating non-Plankian radiation will be discussed. The old phenomenological model of synchrotron emission from internal shocks is currently experiencing a crisis -- it appears to contradict both shock physics and observations. I will describe a different emission mechanism: internal motions in the neutron-proton jet generate electron heat via nuclear and Coulomb collisions, and the electrons radiate the received energy. This mechanism invokes no phenomenological parameters, and the produced radiation can be calculated from first principles. Remarkably, the predicted spectra agree with observations. I will also discuss the central engine and the mechanism driving the jet. The jet may be a strongly magnetized outflow driven by rotation of the central object. Alternatively, the jet may be driven by thermal pressure due to neutrino heating around the central object. Both mechanisms can supply the observed luminosities, but require extreme rotation rates and magnetic fields.