Getting Biblical in Daily Life
Muslim Perspective
If you take a gander at our discussion about Solomon in the "Figures" section, you'll see that Muslim tradition holds that God gave Solomon even greater power than what's outlined in 1st Kings: He can talk to animals and control the wind and jinn (genies). One further difference in the Muslim view of Solomon: he never goes bad. Muslims get to skip the depressing part of Solomon's life, because their tradition states that he remains faithful to Allah 'til the day he dies (Quran, Surah 27).
Elijah, too, holds a place of great respect in Islam. Known as Ilyas, he is not only a faithful prophet of Allah in the Quran, but he also shows up as a sort of guardian angel-type character in many Islamic folk stories (source).
Jewish Perspective
Among other Jewish rituals, Elijah traditionally plays a role in circumcision and Passover ceremonies. This is sort of like a punishment for Elijah. He wrongly accused all of Israel of breaking its covenant with God (19:14, 18), so now he has to watch every time modern Jews don't break their covenant (source). Seems harsh, but if anybody can handle it, it's Elijah. And at least at the Passover feast (Seder), he's treated as an honored guest. Traditionally the host opens the door to invite Elijah into the feast, and a cup of wine is poured just for him (source).
Christian Perspective
Elijah shows up quite a bit in the New Testament. An angel told the father of John the Baptist that his son would have "the spirit and power of Elijah" (Luke 1:17), and when he grew up some people thought he actually was Elijah (see John 1:21). Since Elijah never died, and his return was prophesied in Malachi 4:5, the Jews thought he could pop up any time. Because Jesus Christ's miracles were so similar (this might be a good time to meander over to Elijah in the "Figures" section) to Elijah's, some people also thought that he was Elijah (see Mark 8:27-28). And at Jesus's transfiguration, Elijah appeared (along with Moses, who probably hung out with Elijah all the time in heaven) and spoke with Jesus (see Mark 9:2-4). Although we don't know what Elijah said, we're pretty sure it was awesome.
Catholic Perspective
The Carmelites (named after Mount Carmel) are a Catholic order that finds particular inspiration from Elijah. From his example on Mount Horeb, they "learn to listen for the voice of God in the unexpected and in silence. We seek to allow the Word of God to shape our minds and our hearts so that the way we live […] may be prophetic and therefore faithful to the memory of our father Elijah" (source). In essence, they want to be just like Elijah. Who doesn't?
Mormon Perspective
Mormons find a lot to think about in 1st Kings. They build temples that, to them, are as sacred as Solomon's temple is in the Hebrew Bible (source). And although the mainstream Mormon faith (formally called the Church of Jesus Christ of Latter-day Saints) officially pulled the plug on polygamy in 1890, it still believes that "At various times, the Lord has commanded His people to practice plural marriage [polygamy]. For example, He gave this command to […] Solomon" (source), though they still think Solomon's marriages to idolatrous women were a bad idea (source).
Mormons also, like most (or all) of Christianity, believe that Elijah was a great prophet, and he shows up even in their most recent religious history. They believe that the prophecy of Elijah's return—"Lo, I will send you the prophet Elijah before the great and terrible day of the Lord comes" (Malachi 4:5)—was fulfilled on April 3rd, 1836 when he appeared to Joseph Smith (and others) in a Mormon temple in Ohio (source).