Although strictly meanwhile more than 99 years old, a well-known example here are the Julia- and Fatou sets. -- These sets were first investigated by Gaston Julia and Pierre Fatou in 1917/18, but this work was more-or-less ignored until the invention of computers made it possible to explore the beauties of these sets, and -- following the works of Benoit Mandelbrot -- they became widely popularized in the 1980's.