We consider the construction of adaptive controllers for minimum phase linear systems that achieve nonzero robustness margins in the sense of the gap metric. The gap perturbation margin may be more constrained for larger disturbances and for larger parametric uncertainties. Working in both L-2 and L-infinity settings, and within the framework of the nonlinear gap metric, universal adaptive controllers are first given achieving stabilization for first-order nominal plants, and the results are then generalized to relative degree one nominal plants. A notion of a semiuniversal control design is introduced, which is the property that a bound on performance exists that is independent of the a priori known uncertainty level, and a characterization is given for when semiuniversal designs outperform the class of memoryless controllers and the class of linear time-invariant controllers. Robust semiuniversal adaptive control designs are given for nominal plants under the classical assumptions of adaptive control in both the L-2 and L-infinity settings. The results are applied throughout to explicit classes of unmodeled dynamics including the Rohrs example.