The follow-the-leader (FTL) gait is an effective walking algorithm for a legged system to traverse a rough terrain. In an FTL gait, all the legs simply place at the footprints made by the legs ahead of them. By this way the demand on foothold selection is significantly reduced. A special category of FTL gaits, called continuous FTL gaits, provide a smooth body motion during walking and enable the legged system to reach a higher speed. In this paper, a comprehensive study of continuous FTL gaits is presented. The equations for two types of continuous FTL gaits are formulated. The stability of these continuous FTL gaits is studied analytically and verified numerically. Strategies of forbidden area avoidance and special methods of large foot adjustment are introduced. The motion resulting from the use of these strategies and methods is simulated and checked using computer graphics.