TITLE: Largest "leap-to-generality" in math history?
QUESTION [48 upvotes]: 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.

REPLY [2 votes]: I want to highlight the Aristotelian logic. 
In human language there was discovered the most general mathematical system which even today forms the basis of science and especially mathematics. From tautologies, combinations of sentences which independently of the context seems to be true, truly non-trivial systems of logical deductions was evolved.