In this paper, we develop two discrete-time strong approximation schemes for solving stochastic differential systems with memory: strong Euler-Maruyama schemes for stochastic delay differential equations (SDDE's) and stochastic functional differential equations (SFDE's) with continuous memory, and a strong Milstein scheme for SDDE's. The convergence orders of the Euler-Maruyama and Milstein schemes are 0.5 and 1 respectively. In order to establish the Milstein scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the sfde's, the use of anticipating calculus methods appears to be novel.