Both the large turbulence structures and the fine scale turbulence of the flows of supersonic jets are sources of turbulent mixing noise. At moderately high supersonic Mach numbers especially for hot jets, the dominant part of the noise is generated directly by the large turbulence structures. The large turbulence structures propagate downstream at supersonic velocities relative to the ambient sound speed. They generate strong Mach wave radiation analogous to a supersonically travelling wavy wall. A stochastic instability wave model theory of the large turbulence structures and noise of supersonic jets has recently been developed. The theory can predict both the spectrum and directivity of the dominant part of supersonic jet noise up to a multiplicative empirical constant. Calculated results agree well with measurements.