Big Oil is redrawing the energy map.

For decades, its main stomping grounds were in the developing world—exotic locales like the Persian Gulf and the desert sands of North Africa, the Niger Delta and the Caspian Sea. But in recent years, that geographical focus has undergone a radical change. Western energy giants are increasingly hunting for supplies in rich, developed countries—a shift that could have profound implications for the industry, global politics and consumers.

Driving...