In this paper, the filter-and-forward (FF) relay design for orthogonal frequency-division multiplexing (OFDM) transmission systems is considered for improving system performance over simple amplify-and-forward (AF) relaying. Unlike conventional OFDM relays performing OFDM demodulation and remodulation, to reduce processing complexity, the proposed FF relay directly filters the incoming signal in the time domain with a finite impulse response (FIR) and forwards the filtered signal to the destination. Three design criteria are considered for optimizing the relay filter. The first criterion is the minimization of the relay transmit power subject to per-subcarrier signal-to-noise ratio (SNR) constraints, the second criterion is the maximization of the worst subcarrier channel SNR subject to source and relay transmit power constraints, and the third criterion is the maximization of the data rate subject to source and relay transmit power constraints. It is shown that the first problem reduces to a semi-definite programming (SDP) problem by semi-definite relaxation (SDR), and the solution to the relaxed SDP problem has rank one under a mild condition. For the latter two problems, the problem of joint source power allocation and relay filter design is considered, and an efficient algorithm is proposed for each problem based on alternating optimization and the projected gradient method (PGM). Numerical results show that the proposed FF relay significantly outperforms simple AF relays with an insignificant increase in complexity. Thus, the proposed FF relay provides a practical alternative to the AF relaying scheme for OFDM transmission.