The paper presents the results of an asymptotic theory of axially symmetric cavity flows at small and zero cavitation number. The results have been obtained on the basis of the variational Riabouchinsky principle and the asymptotic theory of slender bodies. This variational-asymptotic approach has been applied to deduce asymptotic expansions for the shape of the cavity and the force exerted on the cavitator at small cavitation number. At the zero cavitation number an integro-differential equation for the shape of the free streamline has been obtained. An exact integral of the equation has been found and a one-parameter family of solutions has been constructed and which has refined earlier asymptotics of Levinson and Gurevich. The equation and asymptotic expansion are independent of the cavitator shape and in this sense are maximally accurate. Any further amendment of the equation of higher order of accuracy would be connected with the shape of the cavitator.