The innovative educational techniques of discovery and inquiry learning and their relevance to the education of academically gifted children are discussed. Following discussion of various definitions of discovery and inquiry, the two terms are dealt with synonomously to avoid confusion. Brief historical background information is given on the growth of discovery and inquiry as educational techniques. Examined are the relationship of these teaching approaches to theory and also the rationale for implementing such techniques in the education of the gifted. Application of the methods in the classroom is considered in terms of the characteristics which must be present in the classroom and learning environment, tools and materials which facilitate the approach, and teacher functions. Recent research concerning the two teaching methods is reviewed and listed are some current programs that specifically use the discovery-inquiry approach in their methodology. (KW)