The Voyager 1 and 2 spacecraft observed low-frequency radio emissions near 2 and 3 kHz during the interval 1983--1987 while at heliocentric distances from 15 to 27 AU and 11 to 20 AU, respectively. We consider the detailed theoretical and observational requirements for this radiation to be produced near multiples of the plasma frequency fp by nonlinear, weak turbulence, wave-wave processes involving electrostatic Langmuir waves. Constraints on the emission processes and source characteristics are discussed. The minimum brightness temperature of the radiation, 3¿1014 K, requires the Langmuir waves to be generated by a plasma instability, most plausibly by an electron beam instability. This requires electron acceleration and beam production in the near vicinity of the source. Minimum Langmuir wave electric fields in the source region, based solely on the kinematics of the radiation processes, lie in the range 1--100 &mgr;V m-1 for nominal source and electron beam parameters. Path lengths for fundamental and harmonic emission processes are calculated: the observed levels of radiation can be produced in path lengths (and source dimensions) smaller than 1 AU provided the participating Langmuir waves have effective temperatures TL~1017 K. These Langmuir wave levels are plausible, based on observations in planetary foreshocks and theoretical calculations. Previous source models for the radiation are discussed. Suggestions that the radiation is generated on the downstream side of the inner heliospheric shock or in the vicinity of the heliopause are shown to face severe theoretical and observational problems, based on present knowledge of the outer heliosphere. However, the upstream (sunward) side of the inner heliospheric shock is predicted theoretically to have a foreshock region containing electron beams and associated Langmuir waves. Theoretically this source region can plausibly produce radiation at multiples of fp with the observed brightness temperatures. ¿ American Geophysical Union 1992