The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optimal rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp (--r:s)/(Ts + I). Numerical experience indicates that L 00 optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L 00 approximation errors than Hankel optimal rational approximations. Pade approximations here exhibit an optimal rate of L 00 approximation, but may show very poor L 00 error behaviour in low-order rational approximations.