The problem of output regulation for general multivariable linear systems has been solved in the 70s, in the seminal works of Francis, Wonham and Davison, under the assumption that the reference signals and the disturbances acting on the system are generated by a known exogenous linear system (the exosystem). The regulator is designed to embed an internal model of the exosystem, which ensures that asymptotic regulation is maintained under arbitrary plant perturbations that do not destroy linearity and closed-loop stability. This robustness property, however, is inexorably lost whenever the internal model does not match exactly the exosystem. In this paper we endow the linear regulator with a discrete-time adaptive unit that adapts the regulator's internal model on the basis of the closed-loop evolution. Compared to existing approaches, adaptation here is cast as an identification problem, and the corresponding optimal predictor is designed independently from the underlying control system. This permits to separate stabilization and adaptation, thus naturally handling general non-square multivariable non minimum-phase plants. Closed-loop stability is guaranteed and, if the dimension of the internal model is large enough and a persistency of excitation condition is fulfilled, asymptotic regulation is achieved for references and disturbances generated by an unknown exosystem. Robustness to parametric uncertainties is inherited by the linear regulator and robustness to additional unmodeled disturbances is proved to hold. (C) 2019 Elsevier B.V. All rights reserved.