This most sacred of Israelite relics was lost at some point between the days of Solomon and Ezra, though we don't know when, where or how (Exodus 25:10-22).
The Kebra Nagast (an account written about the origins of the "Solomonic dynasty" Solomonic line of the "Emperor of Ethiopia" Emperors of"Ethiopia" Ethiopia and also called the Book of the Glory of Kings) prominently mentions the Ark of the Covenant.
According to the Kebra Nagast, to safeguard the Ark from Solomon's growing apostasy, it was secretly taken to Ethiopia, leaving behind a replica that faithful priests had been asked to make. While this sounds rather unlikely, it is nevertheless widely believed among Ethiopians today that their nation is in actual possession of the Ark of the Covenant—that it sits guarded and unapproachable in an old church in the city of Aksum in northern Ethiopia. In fact, each local church in Ethiopia has its own Tabot, or representation of the ark, to memorialize that conviction.
British journalist Graham Hancock, in his book The Sign and the Seal: The Quest for the Lost Ark of the Covenant, 1992, actually gives a more plausible explanation, different from the Kebra Nagast, as to how the ark might actually have ended up in Ethiopia. He speculates that the ark was taken out of Judah by the Levites to protect it from the apostasy of Hezekiah's son Manasseh—that when Josiah later told the Levites to put the ark back into the temple (2 Chronicles 35:3) this was never done, as it had supposedly already been moved to a new temple at a Jewish colony in Aswan in southern Egypt.
Historically, these Jewish colonists were later forced to flee from the Egyptians, and Hancock provides some evidence that they migrated south into Ethiopia—with, he maintains, the Ark of the Covenant. This hypothesis is also explored in a 2002 book titled In Search of the Lost Ark of the Covenant by Robert Cornuke and David Halbrook. Author Grant Jeffrey, in Armageddon: Appointment with Destiny, 1990, while embracing the Kebra Nagast version of events, lends some support to the ark's residing in Ethiopia today (pp. 108-122, 229-233).
Still, there are other theories about the ark's whereabouts that also appear credible—including the possibility that Jeremiah hid it or took it with him at the time of the Babylonian destruction of Jerusalem. The apocryphal book of 2 Maccabees (2:1-8) says he hid it in a cave on Mount Nebo. (Realize, however, that while the apocryphal books can be useful historical sources like many other secular writings, they are not inspired Scripture and often contain errors.)
Many others believe the ark was hidden in a chamber under the Temple Mount. There is, of course, also a very strong possibility that God allowed it to be destroyed by the Babylonians -- when King Nebuchadnezzar captured and destroyed Jerusalem and the Temple in which the Ark rested.
While all these accounts are interesting, there is simply no way to be sure where the Ark of the Covenant may be located. There is also the distinct possibility that, at some time in the past, God allowed it to be destroyed.
Finally, while such matters are certainly interesting, we should avoid getting caught up in them to the exclusion of more important spiritual study.