We propose a distance measure between two probability distributions, which allows one to bound the amount of belief change that occurs when moving from one distribution to another. We contrast the proposed measure with some well known measures, including KL-divergence, showing how they fail to be the basis for bounding belief change as is done using the proposed measure. We then present two practical applications of the proposed distance measure: sensitivity analysis in belief networks and probabilistic belief revision. We show how the distance measure can be easily computed in these applications, and then use it to bound global belief changes that result from either the perturbation of local conditional beliefs or the accommodation of soft evidence. Finally, we show that two well known techniques in sensitivity analysis and belief revision correspond to the minimization of our proposed distance measure and, hence, can be shown to be optimal from that viewpoint.