High mobility channel materials such as Ge and compound semiconductors (CS) show promise for future generation MOSFETs. The challenge is to integrate these materials with a Si substrate and create good interfaces in the devices. Here we show dislocation-free CSOI and Ge-on-insulator (GOI) devices with good characteristics. The InAlAs/InGaAs/InAlAs-OI on Si MESFETs shows a mobility of 8100 cm2/V s. To reduce the leakage current an Al2O3/InGaAs MOSFET was fabricated. Good 451 cm2/V s mobility was obtained, higher than the 340 cm2/V s of GOI MOSFETs. However the marginally better mobility than GOI and 18X lower mobility than MESFETs indicate that the soft phonon scattering, high-κ interface scattering and process variations are challenges for CS MOSFETs. In contrast, the GOI CMOS provides a simpler process and significantly higher electron and hole mobilities than its Si counterparts.