Three control system design problems aimed at meeting specifications given in terms of the l\sb∞ norm, or peak magnitude, of disturbances and errors are solved. The specification in the first problem requires simply that the error satisfy an l\sb∞ norm constraint for all time provided that the disturbance satisfies a corresponding constraint. This is the standard l\sb1 problem, and the results here include generalizations of many known results on this problem: existence of optimal compensators, FIR sub-optimal approximation, and super-optimal approximation. The specifications in the remaining two problems are based on l\sb∞ measures of weighted disturbances and errors; these weights can be chosen such that constraining a weighted signal constrains, for example, its peak rate and/or acceleration. One is an incremental weighted specification, requiring that the weighted error satisfy an l\sb∞ norm constraint up until any given time provided that the weighted disturbance satisfies a corresponding constraint up until the same time. The other is a weighted specification which requires that the weighted error satisfy an l\sb∞ norm constraint for all time provided that the weighted disturbance satisfies a corresponding constraint for all time. The associated design problems turn out to be distinct. For each of the two weighted specifications an appropriate system norm (or gain with respect to the given weights) is defined and it is shown that it can be computed by solving a standard l\sb1 problem (in the incremental weighted case) or a very similar problem (in the weighted case). Results for each weighted design problem parallel those for the unweighted, or standard l\sb1, case: existence of optimal compensators, FIR sub-optimal approximation, and super-optimal approximation.