We study the well-known lion-and-man game in which a lion (the pursuer) tries to capture a man (the evader). The players have equal speeds and they can observe each other at all times. While the game is well-studied in two dimensional domains such as polygons, very little is known about its properties in higher dimensions. In this paper, we study the lion and man game on the surface of convex terrains. We show that the lion can capture the man in a finite number of steps which is a function of the terrain geometry.