TITLE: Blackbox Theorems
QUESTION [83 upvotes]: By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite Simple Groups (assuming the proof is complete). I think very few people really know the nuts and bolts of the proof but it is widely applied in many areas of mathematics. I would prefer not to include as a blackbox theorem exotic counterexamples because they are not usually applied in the same sense as the Classification of Finite Simple Groups. 
I am curious to compile a list of such blackbox theorems with the usual CW rules of one example per answer.  
Obviously this is not connected to my research directly so I can understand if this gets closed.

REPLY [10 votes]: Deligne's construction of Galois representations corresponding to modular forms.

REPLY [4 votes]: The Cohen-Structure theorem in commutative algebra (classifying complete local rings in some sense).