Early science fiction (pre-Global Warming days) usually showed domes as space colonies. Various writers put domed cities on Venus, Mars, and the moon. Other writers used domes cities to show the ills of ouor society and government. In the short story by E.M. Foster "The Machine Stops", humans live in a vast series of rooms in an underground machine that provides everything they could ever need. However, the machine eventually breaks down causing the inevitable death of all its citizens. Arthur C. Clarke saw domes as a retreat in The City and the Stars, a modern day (1950's style) Eden.
The first mention of a domed city was in 1881 in the white supremacist fantasy Three Hundred Years Hence by William Hay, besides being full of only white people Hay's domed city is mostly used for agriculture. In 1905 H.G. Wells suggested A Modern Utopia where at least part of cities were covered over in glass. in 1931 Ray Cummings wrote Brigands of the Moon where people lived in small glass dome shelters. James Blish created the idea of a spindizzy in his series Cities in Flight in which cities could take off into space at will. Logan's Run originally published in 1976 built domed cities to escape polution and war. In 1982 Ben Bova published A City of Darkness where domes were playgrounds during the day, and terrifying by night.
But these are all adult books you say. Never fear, domed cities infiltrated children's literature as well. The Tripods series by John Christopher has humans living as slaves beneath alien domes (very similar to Battlefield Earth, which is never read due to its scientology affiliation, but is really quite good) In 2010 A Crack in the Sky by Mark Peter Hughes was published with the domes being run by a corporation that isn't admitting that the domes may be falling apart piece by piece. Some may even argue that Incarceron by Catherine Fisher could be considered a "domed" city. Songs of Power by Hilari Bell is about an underground domed city and Away is a Strange Place to Be by H.M. Hoover features space habitats, very similar to domed cities.
But just in case you thought this was all some kind of fictional enterprise, feast your eyes on the newest form of domed city from the minds of the Chinese.