Cavitation phenomena in nonlinearly elastic solids have been the subject of extensive investigation in recent years. The impetus for much of these theoretical developments has been supplied by pioneering work of Ball in 1982. Ball investigated a class of bifurcation problems for the equations of nonlinear elasticity which model the appearance of a cavity in the interior of an apparently solid homogensous isotropic elastic sphere or cylinder once a critical external tensile load is attained. This model may also be interpreted in terms of the sudden rapid growth of a pre-existing microvoid. In this paper, we briefly summarize some of the main results obtained to date on radially symmetric cavitation, using the bifurcation model. The paper is a review and a comprehensive list of references is given to original work where details of the analyses may be found.