The paper is devoted to the formulation of the reachability problem for discrete-time dynamical systems with disturbances. The concept of maxmin and minmax forward and backward reach sets is addressed. Invariance of the backward reach set is discussed. The emphasis of the paper is on discrete-time linear systems, for which the ellipsoidal computational method is described. The synthesis of maxmin and minmax closed-loop control for steering the system to a given target set using ellipsoidal backward reach set approximations is explained. The ellipsoidal method covered in the paper is implemented in the Ellipsoidal Toolbox for MATLAB, a popular collection of ellipsoidal calculus routines freely available online. (C) 2011 Elsevier Ltd. All rights reserved.