The first modern curling iron was patented in 1866. The patent was filed by Sir Hiram Stevens Maxim who interestingly was also the man who designed the machine gun.
While Sir Hiram was the first to patent and is thus credited with the invention of the curling iron in modern times, his was by no means the first curling iron to exist in the world. At the Oriental Institute at the University of Chicago, they have in their collections two Bronze-age curling irons from ancient Egypt. The ancient Greeks also were known to use curling tongs heated in an oven to achieve the popular ringlet hair and beard styles seen on their statues.