The growing use of autonomous underwater vehicles and underwater gliders for a variety of applications gives rise to new requirements in the operation of these vehicles. One such important requirement is optimization of energy required for undertaking missions that will enable longer endurance and lower operational costs. Our goal in this thesis is to develop a computationally efficient, and rigorous methodology that can predict energy-optimal paths from among all time-optimal paths to complete an underwater mission. For this, we develop rigorous a new stochastic Dynamically Orthogonal Level Set optimization methodology. In our thesis, after a review of existing path planning methodologies with a focus on energy optimality, we present the background of time-optimal path planning using the level set method. We then lay out the questions that inspired the present thesis, provide the goal of the current work and explain an extension of the time-optimal path planning methodology to the time-optimal path planning in the case of variable nominal engine thrust. We then proceed to state the problem statement formally. Thereafter, we develop the new methodology for solving the optimization problem through stochastic optimization and derive new Dynamically Orthogonal Level Set Field equations. We then carefully present different approaches to handle the non-polynomial non-linearity in the stochastic Level Set Hamilton-Jacobi equations and also discuss the computational efficiency of the algorithm. We then illustrate the inner-workings and nuances of our new stochastic DO level set energy optimal path planning algorithm through two simple, yet important, canonical steady flows that simulate a steady front and a steady eddy. We formulate a double energy-time minimization to obtain a semi-analytical energy optimal path for the steady front crossing test case and compare the results to these of our stochastic DO level set scheme. We then apply our methodology to an idealized ocean simulation using Double Gyre flows, and finally show an application with real ocean data for completing a mission in the Middle Atlantic Bight and New Jersey Shelf/Hudson Canyon region.