In 1934, Gelfond and Schneider independently gave two different solutions to Hilbert's 7th problem on the transcendence of a^b for algebraic a not equal to 0 or 1 and irrational algebraic b. We shall give a historical panorama of the evolution of the problem, of its solutions, and of how the new techniques developed for its solutions laid the foundation for the last half-century of transcendence theory. We shall conclude with an outlook toward exciting future research directions in transcendence and diophantine geometry. This talk is a reprise of talk we gave in Frankfurt commemorating the 100th anniversary of the birth of Theodor Schneider.