This paper formulates energy efficient driving of gasoline and electric powered vehicles as optimal control problems of various complexity. We show minimizing aerodynamic drag can maximize utilization of energy available at the wheel and requires low and constant speeds. By employing optimal control theory we show periods of maximal acceleration, maximal braking, and coasting often accompany constant speed cruising to satisfy boundary conditions on the states (bang-singular-bang optimal control). In the case of gasoline engine vehicles, analytical optimal control derivations show that pulse and glide operation of the engine while cruising can further reduce fuel use (chattering optimal control). For electric vehicles (EV), quadratic rather than linear dependence of energy use on control input results in different eco-driving patterns from gasoline engine vehicles. For EVs, analytical solution to the two point boundary value optimal control problem could be obtained after model simplification which is compared to numerical solution based on a more accurate model. We also evaluate optimal control solution in the presence of state constraints for EVs. Several simulation case studies are presented to showcase the energy efficiency gains with proposed eco driving strategies. (C) 2019 Elsevier Ltd. All rights reserved.