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In the first part, we prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods developed by Cockburn and Shu for the approximation of conservation laws. This inequality implies convergence of the methods for one-dimensional scalar convex conservation laws and is valid for any order and general triangulations with or without limiting. It also implies that the discontinuous Galerkin methods are $L\sp2$ stable. In the second part, we modify and test high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher and Chan. It has been shown by Liu et al. that WENO schemes constructed from the $r\sp{th}$ order (in L$\sp1$ norm) ENO schemes are ($r$ + l)$\sp{th}$ order accurate. We propose a new way of measuring the smoothness of a numerical solution which results in a 5$\sp{th}$ WENO scheme for the case $r$ = 3 (instead of 4$\sp{th}$ with the original smoothness measurement by Liu et al.). This 5$\sp{th}$ order WENO scheme is as fast as the 4$\sp{th}$ order WENO scheme of Liu et al and, both of the two schemes are about twice as fast as the 4$\sp{th}$ order ENO schemes on vector supercomputers and as fast on serial machines (therefore on parallel machines as well). For Euler systems of gas dynamics, we suggest to compute the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the previous WENO schemes (which use the characteristic values to compute weights), and work well for problems which do not contain strong shocks or strong reflected waves. Many numerical tests are presented to demonstrate the remarkable capability of the WENO schemes, especially the 5$\sp{th}$ order WENO scheme using the new smoothness measurement, in resolving-complicated shock structures.