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The zero-sum, perfect information pursuit-evasion differential game is reviewed. The purpose of the thesis is to formulate a method for generating near-optimal closed-loop solutions to these problems. The method is then applied to a number of example problems in order to check its validity. This method deals with solutions in the small and is based on updating the two-point boundary-value problem by use of the neighboring extremal path concept. The two differential game problems examined are a simple motion problem and a rocket problem. Two separate cases were studied for each problem. One was the fixed final time problem and the other was the free final time with a terminal constraint. Analysis of the results obtained, supports the feasibility of this method to provide near-optimal closed-loop solutions to differential game problems.