Analytical and computational techniques are developed to investigate the stability of converging shock Avaves in cylindrical and spherical geometry. The linearized Chester-Chisnell-Whitham (CCW) equations describing the evolution of an arbitrary perturbation about an imploding shock wave in an ideal fluid are solved exactly in the strong-shock limit for a density profile ρ(r) ∼ r−q. All modes are found to be relatively unstable (i.e. the ratio of perturbation amplitude to shock radius diverges as the latter goes to zero), provided that q is not too large. The nonlinear CCW equations are solved numerically for both moderate and strong shocks. The small-amplitude limit agrees with the analytical results, but some forms of perturbation which are stable at small amplitude become unstable in the nonlinear regime. The results are related to the problem of pellet compression in experiments on inertial confinement fusion.