A general theory is developed on the open resonator with rotational symmetric wave-guides which have been used extensively in the electron cyclotron maser (gyrotron). The theoretical analysis is carried unt under the most general conditions, that is the cross-sec-tion radius r(z) is a slowly varying arbitrary function and the resonaor is open. Prperties of the field solutions in the resonator have been discussed, general expressions for the resonant frequencies and Q values have been derived. Calculation results for a concrete resonator are presented.