A computationally efficient error estimator is needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. Efficiency is of special importance for hyperbolic problems where solution features evolve in space and time. We develop an inexpensive a posteriori error estimation technique based on superconvergence. The superconvergence phenomena is described for one- and multi-dimensional problems and for structured and unstructured meshes. We prove that the discretization error of pth-order approximation converges at the downwind boundaries of elements as O(h2p+1) pointwise for one-dimensional problems and on average for higher dimensions. We further prove that the global error converges at O(hp+2) rate at the Radau points of degree p + 1 for one-dimensional problems. In higher dimensions, point-wise superconvergence as well as Radau polynomials seize to exist. However, we demonstrate that average errors converge at a faster O(hp+2) rate over lines (in two dimensions) parallel to the downwind boundary. The estimator requires O(Npk−1) operations to compute relative to O(Npk) solution complexity on a mesh of N elements in k dimensions for one time step. High-order accuracy is desirable, but difficult to achieve for hyperbolic conservation laws with discontinuous solutions. High-order approximations of such problems develop spurious oscillations near discontinuities. One possible solution is to use a limiter - a nonlinear procedure that suppresses oscillations. However, existing limiters reduce the order of accuracy near smooth extrema. By constructing an indicator that distinguishes smooth and “rough” areas, we restrict the use of the limiter to the elements containing discontinuities. As a result, we obtain sharp resolution of shocks as well as achieve the theoretical high-order rate of convergence in smooth regions.