Where do the elves in 'The Lord of the Rings' live?

Answer 1

Elrond's House, or Rivendell was one location. The other was Lothlórien where Galadriel and Celeborn lived. In the Elven Halls of Mirkwood there were elves. There were also some elves at the Grey Havens. After the War of the Ring Legolas brought some elves from Mirkwood and they settled in the fair land of Ithilien. There were folk of Eriador and Lindon, such as Gildor and his crew. Elves resided at the Tower Hills to guard the palantir.

Answer 2

All of the Dwarves that appear in The Hobbit were originally from Erebor, the Lonely Mountain (or their parents were). However, since being evicted, they had wandered as far west as the Blue Mountains, and were common enough through their wandering that Bilbo was familiar with dwarves. This also gave them the reputation for being gypsies of a sort, and they were commonly shunned as beggars.

Originally, the Dwarves of Durin's line (from which all but 3 of the company descend), inhabited Moria and Gundabad until they were driven out by Orcs (or the Balrog known as Durin's Bane). The main host of Durin's Folk lived in the Iron Hills to the northeast of Erebor, under the kindgom of Dain Ironfoot, Thorin's cousin.

Answer 3

Bilbo has two encounters with elves in his adventures. The first is at Rivendale, where Elrond lives. The second is in Mirkwood, the halls of the Elf King.