The authors describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of the author's goals is to apply known estimates derived for elliptic problems to evolution equations. The new technique is applied to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second-order hyperbolic problem. The error is measured with the aid of the \(L^{2}\)-norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented.