From naval operations to ocean science missions, the importance of autonomous vehicles is increasing with the advances in underwater robotics technology. Due to the dynamic and intermittent underwater environment and the physical limitations of autonomous underwater vehicles, feasible and optimal path planning is crucial for autonomous underwater operations. The objective of this thesis is to develop and demonstrate an efficient underwater path planning algorithm based on the level set method. Specifically, the goal is to compute the paths of autonomous vehicles which minimize travel time in the presence of ocean currents. The approach is to either utilize or avoid any type of ocean flows, while allowing for currents that are much larger than the nominal vehicle speed and for three-dimensional currents which vary with time. Existing path planning methods for the fields of ocean science and robotics are first reviewed, and the advantages and disadvantages of each are discussed. The underpinnings of the level set and fast marching methods are then reviewed, including their new extension and application to underwater path planning. Finally, a new feasible and optimal time-dependent underwater path planning algorithm is derived and presented. In order to demonstrate the capabilities of the algorithm, a set of idealized test-cases of increasing complexity are first presented and discussed. A real three-dimensional path planning example, involving strong current conditions, is also illustrated. This example utilizes four-dimensional ocean flows from a realistic ocean prediction system which simulate the ocean response to the passage of a tropical storm in the Middle Atlantic Bight region.