We are concerned with the weak approximation of the weak solutions of stochastic functional differential equations (SFDEs) by means of the Euler–Maruyama schemes. Under quite general assumptions on the coefficients, we provide an estimate for the Lévy-Prokhorov metric between the paths of the weak solution of the original SFDE and the corresponding scheme. The weak convergence rate in terms of the Lévy-Prokhorov metric is characterized by the probability of a "rare event" where the weak solution of the SFDE exits a large sub-domain, which can be easily estimated in many examples. The proof is based on the generalized coupling approach which has been studied in the field of ergodicity of infinite dimensional stochastic systems. We apply our general results to many examples appearing in mathematical finance, economics, physics and so on, to which the existing works can not apply.