Largest "leap-to-generality" in math history? Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that would generalize existing ones and provide a unified, more elegant and more efficient way to think of a class of objects.
What are your favorite examples of such generalizations and their authors ?
(NB: this is a soft question and largely an excuse to commemorate once more the passing of Alexander Grothendieck this week.)
Grothendieck, who is said to have been both very humble and at times very difficult to cope with, reportedly had (source, the quote comes from L. Schwartz's biography) an argument with Jean Dieudonné who blamed him in his young years for "generalizing for the sole sake of generalizing":

Dieudonné, avec l'agressivité (toujours passagère) dont il était capable, lui passa un savon mémorable, arguant qu'on ne devait pas travailler de cette manière, en généralisant pour le plaisir de généraliser. 


Perhaps Claude Shannon deserves a mention for his work, such as defining Entropy in the 1948 paper "A Mathematical Theory of Communication" and generally spurring the field of information theory.  He created very general models for communication and gave some profound results about them.  I don't know much about Shannon's other contributions (besides that they are numerous) so hopefully someone can inform.
Alan Turing's general computing device, the "Turing Machine", could be mentioned.  In a world without electronic computers, "this machine can compute anything that you can" seems to me to be a monumental generalization, even to the foundations of mathematics.
