Long-delay echoes (LDE), defined as echoes received from a fraction of a second to several seconds after a radio signal is transmitted, have been observed off and on for about 50 years. A variety of explanations has been proposed in the past but none is completely satisfactory. The following models are presently proposed for LDE: (1) Radio waves of frequency less than about 4 MHz can become trapped in magnetic field-aligned ionization ducts with L values less than about 4. These waves after being trapped can propagate to the opposite hemisphere of the earth where they become reflected in the topside ionospher. They can then return along the duct, leave it and propagate to the receiver. Delays of up to 0.4 s result and they probably account for most of the LDE at frequencies below 4 MHz with estimated delays of 1--2 s. (2) The signals from two separated transmitters T1 and T2, T2 transmitting a OW or quasi-OW signal, interact nonlinearly in the ionosphere or magnetosphere. If the wave vector and frequency of the forced oscillation at the difference frequency of the two signals satisfies the dispersion relation for electrostatic waves, such a wave would exist and begin to propagate. This wave could grow in amplitude due to wave-particle interaction. At a later time it could interact with the CW signal from T2 and if the wave vector and frequency of the forced oscillation at the difference frequency (frequency of T1) satisfy the dispersion relationship for electromagnetic waves, such a wave would exist and could propagate to T1, or some other receiving station tuned to the frequency of T1. Reasonable ionospheric and magnetospheric plasma parameters lead to delays of up to about 6 s with this model. (3) A large percentage of LDI have been reported with delays of tens of seconds. These delays could be explained if the model in (2) is applied to a magnetospheric ionization duct. Electrostatic waves could propagate for about 1000 km or more over the magnetic equator in such a duct and delays of about 40 s are possible. Dispersion for a finite frequency bandwidth would probably not be so large in cases (1) and (2) as to make voice unrecognizable. Dispersion in model (3) for delays greater than about 10 s would normally be too severe for voice modulation, but occasionally compensating effects might occur for which voice would be recognizable.