Two types of InSb structures, epilayers and quantum wells (QWs), have been grown on on-axis and 6 -off-axis Ge (001) substrates and examined by reflection high-energy electron diffraction, transmission electron microscopy, X-ray diffraction, atomic force microscopy, and the van der Pauw and Hall effect techniques. Anti-phase domain defects, which prevail in these InSb structures when grown on on-axis Ge (001) substrates, are significantly decreased by the use of 6 off-axis Ge (001) substrates. Such off-axis substrates also lead to reductions in the densities of micro-twins and threading dislocations. Room-temperature electron mobilities in 4.0-lm-thick InSb epilayers and 25-nm-thick InSb QWs grown on 6 -off-axis Ge (001) substrates are 59 000 and 14 000 cm 2 /(V s), respectively, which are $1.5 times higher than their counterparts grown on on-axis Ge (001) substrates. These improved mobilities are the highest among the reported values for each type of structure grown on Ge (001) substrates.