Error estimates for fully discretized two- and three-dimensional vortex methods for Euler's equations are given. An elementary proof of the consistency result by \textit{G. H. Cottet} [Thèse de 3ième cycle, to appear], together with its three-dimensional analogue is presented. Following the convergence theory of \textit{J. T. Beale} and \textit{A. Majda} [Math. Comput. 39, 1-27 and 29-52 (1982; Zbl 0488.76024 and Zbl 0488.76025)], the authors show how the stability and consistency results lead directly to the convergence theorems of Beale and Majda. Also algorithms for obtaining vortex methods of high order accuracy in which the computational points are initially distributed on the nodes of nonrectangular grids are described.