It is shown that a general uncertain single-input-single-output regulation problem is solvable only by means of discontinuous control laws, giving rise to the so-called high-order sliding modes. The homogeneity properties of the corresponding controllers yield a number of practically important features. In particular, finite-time convergence is proved, and asymptotic accuracy is calculated in a very general way in the presence of input noises, discrete measurements and switching delays. A robust homogeneous differentiator is included in the control structure thus yielding robust output-feedback controllers with finite-time convergence. It is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high-order sliding-mode controllers. Finally, simulation results are presented. (c) 2005 Elsevier Ltd. All rights reserved.