The phenomenon of Ostwald ripening is examined theoretically for precipitates in solution, crystallites on a substrate, and islands of insoluble surfactant molecules at the liquid-air interface. The classical theory of Ostwald ripening, developed by Lifshitz and Slyozhov and independently by Wagner, employs a continuum approach in which the size distribution is a continuous function of particle size and the conservation equation is formulated as in hydrodynamics with the particle growth rate serving as a velocity in size space. It was derived for the case in which the diffusion fields of the individual particles do not overlap and provides asymptotic expressions at large times for the two extreme cases of diffusion controlled and kinetic controlled growth. In this paper, the validity of this continuum (hydrodynamic) approach is examined using a more rigorous stochastic microscopic approach (MCE). A detailed comparison is made between the numerical and asymptotic results obtained using the Lifshitz, Slyozhov, Wagner theory and the stochastic microscopic continuity equation (MCE). It is shown that with the MCE, the limiting case of "diffusion controlled Ostwald ripening" ceases to have significance, since the kinetics at the interface play a role even when the transport of molecules from the bulk to the particle is diffusion controlled. For the case of interface kinetic controlled growth in precipitates, the asymptotic trends, which are independent of the initial conditions in both cases, are seen much earlier with the MCE. Besides, for this case (precipitates), the MCE predicts a mean radius which is about 25% larger at any given time. While the scaled (with respect to the maximum) asymptotic distributions obtained from the two approaches are similar, the absolute (not scaled) particle size distributions obtained using the MCE are much broader and the maxima are almost a third of those obtained using the Lifshitz, Slyozhov, Wagner approach. Similar differences in the results are obtained for the two dimensional Ostwald ripening of crystallites on a substrate and islands of insoluble surfactant.