This study presents problem solving strategies and processes of thinking of mathematically gifted elementary children with respect to non-routine word problems. The data stem from a university-based course, especially designed to foster gifted children, ages 6-10 years, through the enrichment of the elementary mathematics curriculum. Videotapes of the children's problem solving processes were transcribed in great detail and provided the basis for the analysis. The presented examples show that mathematically gifted elementary students stand out in the ability to work systematically and quickly, getting an insight into the problem's mathematical structure. Additionally these children stand out in their high ability to verbalise and to explain their solutions. In comparison to non-gifted children these qualities in problem solving show significance. The significance was calculated by non-parametric tests. (Contains 5 figures, 4 examples, and 16 footnotes.)