This paper is concerned with the orthogonal frequency division multiplexing (OFDM) systems in high-mobility scenarios, where the wireless channels are modeled as rapidly time-varying (TV) fading channels. To analyze the effect of the mobility on the performance, we derive lower bounds on the capacity of the systems. With Jakes' model, the derived lower bounds depend on both the velocity and the channel state information (CSI). Numerical results reveal two factors that degrade the performance. One is the terminal mobility, and the other is the channel estimation error (tending to be more severe due to the high mobility). When CSI is perfect, the capacity changes very little with the increasing of mobility in the low signal to noise ratio (SNR) region, but has a significant loss in the high SNR region. When CSI is imperfect, improving the accuracy of channel estimation will increase the capacity significantly. We also design block Markov superposition transmission (BMST) codes for the OFDM systems in the high-mobility scenarios. Simulation results coincide with the capacity analysis.