We present a comprehensive investigation of the low-¯eld hole mobility in strained Ge and III-V (GaAs, GaSb, InSb and In1¡xGaxAs) p-channel inversion layers with both SiO2 and high-· insulators. The valence (sub)band structure of Ge and III-V channels, relaxed and under strain (tensile and compressive) is calculated using an effcient self-consistent method based on the six-band k ¢ p perturbation theory. The hole mobility is then computed using the Kubo-Greenwood formalism accounting for non-polar hole-phonon scattering (acoustic and optical), surface roughness scatter- ing, polar phonon scattering (III-Vs only), alloy scattering (alloys only) and remote phonon scattering, accounting for multi-subband dielectric screening. As expected, we find that Ge and III-V semiconductors exhibit a mobility significantly larger than the \universal" Si mobility. This is true for MOS systems with either SiO2 or high-k insulators, although the latter ones are found to degrade the hole mobility compared to SiO2 due to scattering with interfacial optical phonons. In addition, III-Vs are more sensitive to the interfacial optical phonons than Ge due to the existence of the substrate polar phonons. Strain { especially biaxial tensile stress for Ge and biaxial compressive stress for III-Vs (except for GaAs) { is found to have a significant beneficial effect with both SiO2 and HfO2. Among strained p-channels, we find a large enhancement (up to a factor of 10 with respect to Si) of the mobility in the case of uniaxial compressive stress added on a Ge p-channel similarly to the well-known case of Si. InSb exhibits the largest mobility enhancement. In0:7Ga0:3As also exhibits an increased hole mobility compared to Si, although the enhancement is not as large. Finally, our theoretical results are favorably compared with available experimental data for a relaxed Ge p-channel with a HfO2 insulator.