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.

Solution 1:

Surely the step from numbers to groups and fields (which is due mostly to 19th-century mathematicians such as Abel, Galois and Dedekind) must count as one one of the greatest leaps forward in history.

Much of what was already known about numbers was quickly reproven for abstract algebraic structures – and thus for an infinite number of concrete structures with sheer limitless applications.

Solution 2:

To modern eyes the definition of metric space by Frechet in 1906 may not seem much, but it paved the way to modern analysis and was a huge leap forward. The ubiquity of metric spaces in virtually all realms of mathematics also shows how profound Frechet's contribution was.

Solution 3:

My 5 cents in favor of category theory by Samuel Eilenberg and Saunders Mac Lane.

Not only did it provide a unifying language for very diverse groups of objects/relations, it was the first (to my very limited knowledge) abstract theory to put the emphasis on morphisms preserving structure (rather than structures themselves), and it paved the way to Grothendieck's Topos theory.

Also, at first sight it can seem quite over-abstract and pointless (see for instance http://en.wikipedia.org/wiki/Abstract_nonsense), which makes its power even more surprising.

Edit: I also need to include Laurent Schwartz for his theory of distributions, which (at least in common terms) are generalizations of $L^1$ functions and Radon measures. The theory finally gave a proper context to write things like $H' = \delta$ (with $H$ the Heavyside function and $\delta$ the Dirac "function"). Anecdotally, it helped me understand the point of taking the smallest possible set (here test functions) to build the largest possible dual space.