As noted in a recent post, maps of empires tend to exaggerate their territorial extents, and the Achaemenid Persian Empire (550-330 BCE) is no exception. Most maps of this important empire depict it as covering all or almost all the South Caucasus region, with its border extending to the crest of the Greater Caucasus range (see the top maps from a Google image search posted below). Some show it as pushing even further to the north, encompassing the historically Circassian lands to the north and west of the Caucasus and sometimes even extending completely around the Black Sea (see below).
There is little if any good evidence, however, that the Achaemenid Persian Empire ever included the Kingdom of Colchis, located mainly in what is now the western half of the Republic of Georgia. The Wikipedia map of the early Georgian states posted below gives a much better depiction of the geopolitical situation of the time. The notion that this ancient Persian empire extended to the crest of the Greater Caucasus range derives essentially from a passage written by the ancient Greek scholar Herodotus. Although there is much to admire in the works of Herodotus, it has long been known that many of his assertions were far from accurate. It is for good reason that Lloyd Llewellyn Jones recently decided that it was necessary to write a book on the Achaemenid Empire based mostly on Persian sources, rather than on Herodotus and other Greek writers. But Jones, unfortunately, also maps western Georgia as having been under Persian control.
There is, however, some scholarly disagreement about which polity (or polities) had ultimate sovereignty over what is now western Georgia between 550 and 330 BCE. The Wikipedia article on the history of the Republic of Georgia provides an excellent summary:
Between 653 and 333 BC, both Colchis and Iberia survived successive invasions by the Iranian Median empire. The case is different for the Achaemenid Persians, however. According to Herodotus (3.97), Achaemenid power extended as far as the Caucasus mountains, but the Colchians are not included in his list of the twenty Persian satrapies. Nor are they referred to in the lists of Achaemenid lands (dahyāva) given in the Old Persian inscriptions of Darius and his successors. In Xenophon’s Anabasis (7.8.25; probably an interpolation) the tribes of Colchis and East Pontus are referred to as independent (autónomoi). On the other hand, Herodotus mentioned both the Colchians and various Pontic tribes in his catalogue (7.78-79) of approximately fifty-seven peoples who participated in Xerxes’ expedition against Greece in 481-80 BC. As the Encyclopaedia Iranica states, it is thus probable that the Achaemenids never succeeded in asserting effective rule over Colchis, though local tribal leaders seem to have acknowledged some kind of Persian suzerainty. The Encyclopaedia Iranica further states, whereas the adjoining Pontic tribes of the nineteenth satrapy and the Armenians of the thirteenth are mentioned as having paid tribute to Persia, the Colchians and their Caucasian neighbors are not; they had, however, undertaken to send gifts (100 boys and 100 girls) every five years (Herodotus 3.97).
The giving of gifts and the supplying of troops by a polity to a much more powerful neighboring empire, however, does not in itself indicate inclusion in that empire. It must also be noted that careful historical cartographers, such as Thomas Lessman, do not map western Georgia as having been part of the Achaemenid Persian Empire (see the map below). 
The issue at stake here is not merely that of the inaccurate mapping of empires. What I am more concerned about is historical amnesia about the Caucasus, coupled with its pervasive historical misrepresentation. To put it simply, this key region of the world does not get its due in most historical and geographical accounts. All too often, it is simply appended to one or more empires based in other lands. Many such empires did covet the region, and in some periods they did control, directly or indirectly, large parts of it. But the Caucasus also had its own kingdoms and other polities, which deserve recognition.
I recently gave a keynote address about such issues at a conference on the Black Sea region held in Batumi in the Republic of Georgia. I hope to convert this talk to a video later this year; if I do so, I will post it on this website.