In 1961, Robert Mac Dowell and Ernst Specker proved that every model of Peano Arithmetic has an elementary end extension. I will explain what it means and why the result and its proof are important. Until today, the theorem has been a constant source inspiration in the search for the ultimate version of the result. The work involves models of second-order arithmetic, automorphism groups, lattices of elementary substructures, and more. I will survey some recent results.