A universal adaptive controller is constructed that achieves asymptotic tracking of a given class of reference signals and asymptotic rejection of a prescribed set of disturbance signals for a class of multivariable infinite-dimensional systems that are stabilizable by high-gain output feedback. The controller does not require an explicit identification of the system parameters or the injection of a probing signal. In contrast to most of the work in universal adaptive control, this paper is based on an input-output approach and the results do not require a state-space representation of the plant. The abstract input-output results are applied to retarded systems and integrodifferential systems.