The jet obliquity compensation approach for cutting plane curve and space curve with abrasive water jet (AWJ) has been researched. The kerf width, kerf taper and kerf lateral surface roughness of steel and aluminum alloy cut by AWJ have been studied. In order to improve the cutting efficiency and cutting quality, the jet obliquity compensation approach for cutting plane curve and space curve with AWJ and the concept of equivalent cutting parameters have been proposed. The relationships between equivalent cutting parameters in curve proceeding and nominal cutting parameters in linear proceeding have been studied. The mathematical models of nozzle turning radius have been established based on the transverse obliquity and kerf inner-outer radius compensation, and is verified by cutting plane ellipse.The effect of jet pressure, nozzle traverse speed, standoff distance and abrasive flow rate on kerf width and kerf taper has been researched. The range analysis and variance analysis have been made on experiment results, and the regression formulas of kerf width and kerf taper have been established by regression analysis. The kerf width and kerf taper when cutting steel Q235 and aluminum alloy 5052 by AWJ have been studied. It is shown that the standoff distance is the most important impact factor to the kerf width and kerf taper and the kerf width increases with an increment in standoff distance, and the effect of jet pressure, nozzle traverse speed and abrasive flow rate is almost the same. The kerf width cut by abrasiver 100# is wider than that by abrasiver 80#. The kerf width of aluminum alloy 5052 is wider than that of steel Q235 at the condition of same size abrasive. The jet capability for cutting aluminum alloy is stronger than that for steel at the condition of the same abrasive flow rate. The kerf taper decreases with a reduction in standoff distance and the kerf taper cut by abrasiver 100# is greater than that by abrasiver 80# at the same cutting conditions. The kerf lateral surface roughness cut by AWJ has been measured. The results show that the nozzle traverse speed is the most important impact factor to surface roughness and the second impact factor is abrasive flow rate. The kerf lateral surface roughness decreases with a reduction in nozzle traverse speed and an increment in abrasive flow rate. The kerf lateral surface roughness by 80# is greater than that by abrasiver 100# at the same cutting conditions. And the kerf lateral surface roughness of steel Q235 is smaller than that of aluminum alloy 5052.Based on the requirement of controlling the kerf taper and kerf lateral quality, the actual cutting parameters in cutting of jet obliquity compensation have been researched and the concept of equivalent cutting parameters has been proposed. The mathematical models of equivalent cutting parameters have been built up. The semi-experimental models of kerf width and kerf taper when cutting steel Q235 and aluminum alloy 5052 have been established respectively. These models can provide theoretical guidance for active controlling the kerf taper, kerf width and kerf lateral quality by AWJ, and are adequate for cutting plane curve and space curve.By using the semi-experimental models of kerf width and kerf taper, the kerf width and kerf taper can be controlled to improve the cutting quality. The cutting technologies of plane ellipse and saddle curve have been researched. The plane ellipse has been cut on steel Q235 with the abrasiver 80#. The kerf width and kerf taper are compensated by using the approach of jet obliquity compensation. On the basis of the established models of nozzle turning radius based on the transverse obliquity and kerf inner-outer radius compensation, the model of nozzle turning radius based on the transverse obliquity and kerf inner-outer radius compensation for cutting plane ellipse has been set up and validated experimentally. The model of nozzle turning radius based on the transverse obliquity and kerf inner-outer radius compensation for cutting saddle curve has been set up and the programme chart for cutting saddle curve has been designed. The model for cutting saddle curve is adequate for cutting space curve on any random geometry workpiece and improving the kerf quality.