Optimum confidence bounds proposed by Buehler (1957), primarily for functions of the parameters of distributions on finite sample spaces, are investigated in several directions. These investigations include consideration of: questions of existence and construction, order-restricted optimal regions, simultaneous confidence bounds for sets of parametric functions, Bayes adaptations, and strong optimality for monotone likelihood ratio families. They include as well the exploiting of special features of parametric functions that are terminal-event probabilities of event trees and of parametric functions that express the reliability of monotone systems;Reference;Buehler, R. J. 1957. Confidence intervals for the product of two binomial parameters. Journal of the American Statistical Association. 52:482-493.