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This paper investigates achievable information rate and relay precoder design of multiple-input-multiple-output (MIMO) amplify-and-forward (AF) relay networks under imperfect channel state information (CSI) including channel estimation (CE) errors and feedback/feedforward (FB/FF) delay errors, without considering the direct link from source to destination. For the first time, we derive the achievable information rate of AF MIMO relay networks under imperfect CSI for two scenarios: In the first scenario, both the source-relay (S-R) and relay-destination (R-D) links are estimated at the destination, whereas in the second scenario, the S-R link is estimated at the relay, and the R-D link is estimated at the destination. We then design a relay precoder that maximizes the approximate achievable information rate, given the average relay power constraint. Numerical results show that Scenario I achieves better average information rate than Scenario II. It is also observed that the accuracy of the CE of the S-R link is more important than that of the R-D link. Finally, the proposed relay precoder can achieve an average achievable information rate higher than the conventional fixed gain relaying scheme, and higher than a relay precoder design treating the estimated CSI as perfect CSI at higher CE and FB/FF delay errors.