The purpose of this research is to study the mathematical problem solving processes, strategy use and success factors of middle grade gifted and talented (GT) elementary school students. This research is based on 9 mathematical problems edited by the author and divided into the following categories: ¡§numbers and quantity,¡¨ ¡§shape and space,¡¨ and ¡§logical thinking.¡¨ Seven GT students from Ta-Tung elementary school in Kaohsiung were selected as target students in the study. Besides, the seven students were translated into original cases using a thinking aloud method. Here are the conclusions: First of all, when facing non-traditional problems, GT students may use different problem solving steps to solve different problems and may not show all detailed steps for every single problem. The same types of problems may not have the same problem solving steps. Missing any single step would have no impact on the answers. Problem solving sequence may not fully follow the traditional 5-step sequence: study the problem, analyze, plan, execute, and verify, and, instead, may dynamically adjust the steps according to the thinking. Secondly, GT students¡¦ problem solving strategy includes more or less the following 19 methods: trial and error, tabling, looking for all possibilities, a combination of numbers, listing all possible answers, classifying the length of each side, classifying graphics, classifying points, adding extra numbers (the triangle problem), drawing, identifying rules and repetition, summarizing, forward solving, backward solving, remainder theory, polynomials, organizing data, direct solving, and making tallies. Finally, problem solving success factors are tightly coupled with problem solving knowledge, mathematical capability, and problem solving behavior. Problem solving knowledge includes knowledge of language, understanding, basic models, strategy use, and procedural knowledge. Instances of mathematical capability are capability of abstraction, generalization, calculation, logical thinking, express thinking, reverse thinking, dynamic thinking, memorizing, and space concept. Problem solving behavior includes the sense of understanding the problem and mathematical structure, keeping track of all possible pre-conditions, good understanding of the relationship between the problems and the objectives, applying related knowledge or formulas, verifying the accuracy of the answers, and resilience for problem solving. In addition to discussing the research results, future directions and recommendations for teaching mathematics for GT and regular students are highlighted.