This thesis explores and develops kinematics modeling and stability analysis of hybrid robots and their most suitable mode of operation for a given terrain. Hybrid robots combine two or more types or modes of operation, or such as rolling on all wheels, walking on feet, and combination of wheels and feet. Each mode of operation is suitable for a particular type of terrain, i.e. rolling on wheels is most desirable for relatively flat surfaces, while walking is suited to rough terrain, and walking on front leg and rolling on back wheels, called pulling, is suitable for relatively smooth but slippery terrain. The basis the development of the kinematic modeling is on the fundamental work of Tarokh et al. This thesis specializes the modeling to a particular robot which is designed in this thesis for operation in different modes. Each mode is analyzed in terms of stability and efficiency for a terrain. In particular an efficiency measure is developed here for each mode in order to decide the optimal mode of operation. As part of this research a program is developed to model and simulate the particular versatile hybrid robot and the simulation results are presented in this thesis that demonstrates the methodologies. The design of the 3-mode hybrid robot, interaction of the robot with the terrain in different and challenging terrains, and the optimal choice of operation mode are the main, and believed to be the original, contributions of this thesis.