Methods are given for the numerical computation of Shannon's rate-distortion functionR(D)for certain memoryless message sources. It is first assumed thatU, the set of possible message-source outputs, andV, the set of possible destination symbols, are countable. The computation ofR(D)for this case is reduced to a minimization problem in which the variables are the destination-symbol probabilities. For arbitraryUandV, upper and lower bounds onR(D)are derived by partitioningUandVeach into a countable collection of disjoint subsets and employing the results derived previously for the case of countableUandV. Conditions are then discussed under which these bounds can be made arbitrarily close to each other by choosing sufficiently fine partitions ofUandV. Two examples are included to illustrate the results in detail.