The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't even listed in its index. The Wikipedia article on isomorphisms only gives two concrete examples.
There are many surprising, significant, classic isomorphisms. I'll refrain from giving examples. What are your favorites?
As usual, please limit yourself to one isomorphism per answer.
(Related: your favorite surprising connections in mathematics. But this question is looking for more concrete examples, particularly those that illustrate the power of the idea.)