In the recent times I have heard a lot about the following:

1. The Atiyah-Singer Index theorem
2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications. 

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.

Edit: With my thin background in geometry and topology I cannot make the question more precise or add more examples but I can tell why I would like to know such programs. After I heard these two theorems and their applications I thought every mathematician should know a little about these results even if they don't try to completely understand its proof. For example, Hamilton's 1982 result mentioned by PVAL is something I want to read about. Thanks for many good answers. Again, I emphasize that I have a very thin background in these things.