Böhm-Jacopini theorem
I had never heard about it before, but taking a quick look at it I can understand why it is considered so important.

Basically, being able to prove that any program can be implemented via three basic operations (sequence, seletion, iteration) allows for simpler formalization when designing a new programming language.

It's basically a one-to-one correspondance to a Turing machine programme, therefore you can easily prove your programming language to be Turing complete.