What giftedness really means has been discussed over centuries. This exploration, by a researcher involved in the study of mathematically talented youth, considers that giftedness may take many forms. The construct of general intelligence is probably the most widely studied psychological construct, but it is apparent that the IQ is not an ideal measure to use in grouping school children for instruction in specific subjects, since the IQ is the aggregate of many cognitive abilities. The Study of Mathematically Precocious Youth at Johns Hopkins University is one research project that has attempted to define giftedness in its several domains. Mathematically talented students have been identified at early ages through achievement tests. However, specialists in gifted children research are beginning to look at more subjective ways of evaluating giftedness, consistent with theories of multiple intelligences. In fact, all that can be done for gifted children, however defined, is to provide them the special, supplemental, and perhaps accelerative educational experiences appropriate to their abilities and interests. Commentaries by Howard Gardner ("Six After Thoughts: Comments on J. Stanley, 'Varieties of Giftedness'") and Joyce VanTassel-Baska ("Response to Julian Stanley's Invited Address at AERA Entitled 'Varieties of Giftedness'") are attached. (Contains 3 tables, 1 figure, and 89 references.) (SLD)