The dispersion equation of a coaxial arbitrary-shaped-groove slow-wave structure is derived by means of an approximate field-theory analysis，in which the continuous profile of the groove is approximately replaced by a series of recta ngular steps，and the field continuity at the interface of two neighboring step s and the matching conditions at the interface between the groove region and central region are employed.The simulation results by CST MWS are in good agreement with the numerical calculation results of the dispersion equation.We have calculated the dispersion characteristics and the coupling impedance of the slo w-wave structures with some special groove shape.It shows that the dispersion c haracteristic of the triangle-groove structure is the weakest and the coupling i mpedance of it is the least, while the dispersion characteristic of the inverted -trapezoid-groove structure is the strongest and the coupling impedance of it is the largest.