Jude Law's character asks to be called by his middle name, Eugene. "Eugene" comes from the Greek for "well born," which he is. "Eugenics" (the science of improving the hereditary qualities of a race or breed) is the central theme of the film.
Uma Thurman and Ethan Hawke became a couple during the filming of this movie. The two later married in 1998, and had two children, but divorced in 2005.
When Gattaca was first released, as part of a marketing campaign there were adverts for people to call and have their children genetically engineered. Thousands of people called, wanting to have their offspring genetically engineered.
Many of the "futuristic" buildings in the film are actually quite old. Many of these represent a type of postmodern architecture called "brutalism", which was popular in the 1950s. The two massive arches seen behind Jerome and Irene during their talk are actually the spillway of the Sepulveda Dam in Los Angeles, which was built in the 1930s.