Movable antenna (MA) has been deemed as a promising technology to flexibly reconfigure wireless channels by adjusting the antenna positions in a given local region. In this paper, we investigate the application of the MA technology in both decode-and-forward (DF) and amplify-and-forward (AF) relaying systems, where a relay is equipped with multiple MAs to assist in the data transmission between two single-antenna nodes. For the DF relaying system, our objective is to maximize the achievable rate at the destination by jointly optimizing the positions of the MAs in two stages for receiving signals from the source and transmitting signals to the destination, respectively. To drive essential insights, we first derive a closed-form upper bound on the maximum achievable rate of the DF relaying system. Then, a low-complexity algorithm based on projected gradient ascent (PGA) and alternating optimization (AO) is proposed to solve the antenna position optimization problem. For the AF relaying system, our objective is to maximize the achievable rate by jointly optimizing the two-stage MA positions as well as the AF beamforming matrix at the relay, which results in a more challenging optimization problem due to the intricate coupling variables. To tackle this challenge, we first reveal the hidden separability among the antenna position optimization in the two stages and the beamforming optimization. Based on such separability, we derive a closed-form upper bound on the maximum achievable rate of the AF relaying system and propose a low-complexity algorithm to obtain a high-quality suboptimal solution to the considered problem. Simulation results validate the efficacy of our theoretical analysis and demonstrate the superiority of the MA-enhanced relaying systems to the conventional relaying systems with fixed-position antennas (FPAs) and other benchmark schemes.