Propagation and localization of cylindrical wave in a two-dimensional isotropic and homogeneous random medium is studied. By expanding the random permittivity fluctuation in the form of a Wiener integral equation in the frequency domain, and representing the wave fields by a linear combination of outgoing and incoming waves, the scalar Helmholtz equation is solved by means of stochastic functional approach to obtain the analytical expression of cylindrical wave. The spatial wave energy distribution is derived to demonstrate the localization phenomenon, and the localization length is also calculated. Compared with the waves in non-random medium, the wave transfer equation between plane wave and cylindrical wave in random medium shows an additional exponential factor to indicate the modulation effects due to the medium randomness in both the amplitude and the phase. Numerical simulations are presented to illustrate the functional dependence of the localization phenomenon.