In this thesis, we study the problem of assembly line balancing with stochastic task process times. The research considers both the well-known straight line balancing problem and U-line balancing problem where the line is paced, with no buffer inventories between stations. The objective is to minimize a two component cost function where the cost terms come from cost of manning the line and cost of finishing the incomplete units off the line. Cost is measured by an existing exact method for straight line balancing and a heuristic cost measurement method is developed for U-line balancing. The key idea in the core of this research is a task's marginal desirability for assignment at a given station. This idea is embedded in a beam search heuristic for solving both the straight line and U-line balancing problem. Extensive computational experiments and simulation experiments are made with well-known problems in the literature under the assumption of normally distributed task processing times. The quality of the solutions found by beam search for the straight-line balancing problem is compared to an existing method in literature. A simulation model of the assembly design is constructed and sample results from the U-line balancing problem are tested against the simulation results. The algorithm presented in this thesis improves the objective function by up to 24 percent.