Adaptive Optics (AO) systems enable to compensate the adverse effects of atmospheric turbulence on ground-based telescopes' images in real time, using a deformable mirror (DM) inserted in the telescope's optical path, and measurements provided by a wavefront sensor (WFS). This paper revisits minimum-variance (MV) control design for astronomical AO systems in a state-space framework. It presents a survey of the modeling and control issues arising in this multi-variable disturbance rejection problem. In a linear time-invariant framework, and under some mild assumptions, the optimal solution to MV control for AO systems is shown to be a discrete-time LQG controller. This result holds for a DM with instantaneous response, and for a fairly general class of DM's dynamics. The state-space approach is extended to Wide-field Adaptive Optics (WfAO) configurations involving several DMs and/or WFSs. Integral-action control used in existing AO systems is compared with the LQG controller. Experimental WfAO results obtained on a laboratory test bench are presented, showing significant improvement in performance. Finally, open issues and perspectives of applicative and/or theoretical interests are discussed. (C) 2012 Elsevier Ltd. All rights reserved.