A critical review is given of the properties of magnetoelastic delay lines, employing yttrium iron garnet (YIG) and having physically distinct input and output ports. Attention is directed to two configurations involving the materials YIG and yttrium alumium garnet (YAG), namely the YAG(nλB/4)-YIG-YAG(n/4) employing YAG quarter-wave plates λB/4, and the YIG-YAG-YIG structures. These offer the highest isolation between input and output ports coupled with the lowest insertion loss for magnetoelastic waves, under specified conditions of microwave frequency and delay. The transfer functions through the YAG material are derived in each configuration for the cases of 1) imperfect matching between the acoustic characteristic impedances of YIG and YAG, with perfect optical bonding, and 2) imperfect optical bonding. Conventional matched filter theory is used to evaluate the time response of each magnetoelastic delay line, linear variation of delay with frequency, to a linearly frequency-modulated pulse. It is established that the configuration employing quarter-wave plates gives desirable weighting of the main pulse. However, a spurious echo of unacceptable level exists, which is delayed approximately by an acoustic round trip in the nλB/4 plate, when a reasonable bandwidth is employed. Analysis shows that this difficulty does not arise for the YIG-YAG-YIG configuration. However, an external weighting network is required for range sidelobe reduction.