Motion of a fluid interface in the Richtmyer-Meshkov instability in cylindrical geometry is examined analytically and numerically. Nonlinear stability analysis is performed in order to clarify the dependence of growth rates of a bubble and spike on the Atwood number and mode number n involved in the initial perturbations. We discuss differences of weakly and fully nonlinear evolution in cylindrical geometry from that in planar geometry. It is shown that the analytical growth rates coincide well with the numerical ones up to the neighborhood of the break down of numerical computations. Long-time behavior of the fluid interface as a vortex sheet is numerically investigated by using the vortex method and the roll up of the vortex sheet is discussed for different Atwood numbers. The temporal evolution of the curvature of a bubble and spike for several mode numbers is investigated and presented that the curvature of spikes is always larger than that of bubbles. The circulation and the strength of the vortex sheet at the fully nonlinear stage are discussed, and it is shown that their behavior is different for the cases that the inner fluid is heavier than the outer one and vice versa.