The Asii or Asioi, were one of the Indo-European tribes mentioned in Roman and Greek accounts as responsible for the downfall of the state of Bactria circa 140 BCE. They have been identified as the Scythians, the Wusun, the Yuezhi, the Tocharians and/or the Alans.
The texts relating to the Asii are very brief. The three main surviving classical sources are those of Strabo, Trogus and Justin. Both Trogus' Historiae Philippicae (as preserved in Justin) and Strabo's Geography exist in a number of ancient manuscripts containing significant textual variations leading to widely varying translations and interpretations.
Trogus wrote his Historiae Philppicae in Latin. Only his 'Prologues' have survived intact. He mentions three tribes involved in the conquest of Bactria: the Asiani, Sacaraucae and the Tochari, of whom the Sacaraucae were said to have been destroyed. The Asiani are reported as becoming, at some point, rulers over the Tochari, though this text is sometimes translated as the "Asian kings of the Tochari."
Marcus Junianus Justinus, a late 2nd or 3rd century Roman historian, wrote an epitome or condensation of Trogus' history. The last datable event recorded by Justin is the recovery of the Roman standards captured by the Parthians in 20 BCE, although Trogus’ original history may have dealt with events into the first decade of the 1st century CE.
Strabo wrote in Greek and completed his Geography in 23 CE, around the time of Trogus. He mentions four tribes: the Asioi (commonly accepted as the equivalent of the Latin Asii), the Pasianoi, the Tacharoi (or Tokharoi) and the Sakaraukai.
Pliny the Elder
Pliny the Elder wrote his Naturalis Historia with a brief mention of a people called the Asini:
Pliny mentions as neighbours of the Soseadae the people of the Asini, who are reigning in the city of Bucephela. From these three data; 1) the Tacoraei are neighbours of the Besadae/Sosaeadae; 2) the Asini are the neighbours of the Sosaeadae; 3) The Asiani are kings of the Thocari, it follows that the Asini of Pliny's text are identical with the Asiani, who are the kings of the Tocharians. This implies that—at least in the time of Pliny—the Kushāṇas were kings of the region between Jhelam and Indus and that Bucephala was one of their cities. It seems that Pliny availed himself of a recent description of this territory and that Ptolemy knew these data too.— 
Theories on the identification of the Asii
Many theories have been proposed by historians and other scholars as to their origins, relationships, language, culture, etc., but so far no consensus has emerged.
It is generally accepted that the Asiani mentioned by Trogus were probably identical to the Asii of Strabo.
There is no agreement over whether another tribe mentioned by Strabo, the "Pasiani" were likewise related. Scholars such as J. Marquart believe that they were synonymous with the Asiani.  W. W. Tarn, Moti Chandra and some other scholars think that "as Asiani is the (Iranian) adjectival form of Asii, so Pasiani would be the similar adjectival form of, and would imply, a name *Pasii or *Pasi". In other words, the Asii and the Pasiani were one and the same, and "Pasiani" was a mispelling of Asiani, or a variant of the same name. Others suggest that the name is a misspelling of Gasiani, a name which is believed by Chinese scholars to be connected to the Kushan Empire (endonym: Kushano; Chinese: Guishuang 貴霜).
Asii or Asiani may simply be a corruption of the name of the Issedones – an Iranian people mentioned by Herodotus – who are frequently identified with the Wusun mentioned in contemporaneous Chinese sources.
Taishan Yu proposes that Asii were "probably" the dominant tribe of a confederacy of four Issedonean tribes "from the time that they had settled in the valleys of the Ili and Chu" who later invaded Sogdiana and Bactria. "This would account for their being called collectively "Issedones" by Herodotus." He also states that the "Issedon Scythia and the Issedon Serica took their names from the Issedones." Yu believes that the Issedones must have migrated to the Ili and Chu valleys, "at the latest towards the end of the 7th century B.C."
It has been suggested that the Wusun may also be identified in Western sources as their name, pronounced then *o-sən or *uo-suən, is not far removed from that of a people known as the Asiani who the writer Pompeius Trogus (1st century BC) informs us were a Scythian tribe.— J. P. Mallory and Victor H.Mair The Tarim Mummies
Yuezhi/Kushans & Tocharians
Alfred von Gutschmid believed that "Pasiani" and other names mentioned by Strabo are an attempt to render Yuezhi in Greek. W. W. Tarn first thought that the Asii were probably one part of the Yuezhi, the other being the Tocharians. However, he later expressed doubts as to this position.
The Asii were identical with the Paisani (Gaisani) and were, therefore, also the Yuezhi.— J. Markwart. Ērānšahr
The Yuezhi and the Wusun were originally two branches of the same people, the Yuezhi being the 'Moon clan'; while the Wusun were the 'Solar clan'.— Yury Aleksey Zuev, Early Turks: Essays of History and Ideology
The Asii were probably one of three Scythian tribes, whereas the Tochari were probably not, and should be identified with the Yuezhi.— A. K. Narain The Indo-Greeks
One of the most important sources of information on nomad migration in Central Asia is Justin's Prologue to Pompeius Trogus (prologue to book XLII), which states that 'the Asiani are kings of the Tochari and destroyed the Sacaraucae' (Reges Tocharorum Asiani interiusque Sakaraucarum). It is possible to conclude from this extract that the Asiani and the Tochari were closely related tribes. What is more, it indicates that the 'Asiani' dominated the 'Tochari' (Reges Tocharorum Asiani). We can identify the Asiani with the Kushans (von Gutschmidt 1888; Haloun 1937; Bachhofer 1941; Daffina 1967), one of the leading tribes, which subsequently came to power and created a great empire. It is noteworthy that Justin says that the Tochari were ruled by the Asiani, while the Chinese sources identify them as the largest of the five Yuezhi principalities.— Kazim Abdullaev, 2007, Nomad Migrations in Central Asia.
The Asii/Asiani have also been identified with the Alans – i.e. a western Central Asian population, rather than the Yuezhi-Tochari of eastern Bactria – from whom the modern Ossetians derive their name.
With this identification of the Asii-Asiani, the Prologues seem instead to concern two later distinct periods already disconnected from the time of Eucratides. Moreover, from a geographical point of view, they describe events not related to the eastern, but to the western border of the Graeco-Bactrian kingdom, that is a region which was in close contact with Parthia. Therefore, the ethnonym of the Asii-Asiani should be transferred westwards, that is to a different historical context (the Kangju area).
- Unesco Staff (January 1, 1994). History of Civilizations of Central Asia. UNESCO. p. 488. ISBN 9231028464. Retrieved February 13, 2015.
- "The Murundas and the ancient trade-route from Taxila to Ujjain." P. H. L. Eggermont. Journal of the Economic and Social History of the Orient, 9 (1966), p. 283.
- Archaeological Reconnaissances in North-western India and South-eastern Īrān, pp. 31–32 and n. 15. Aurel Stein. (1937). Macmillan and Co., London.
- Iaroslav Lebedynsky. (2006). Les Saces: Les «Scythes» d'Asie, VIIIe siècle av. J.-C. — IVe siècle apr. J.-C. Editions Errance, Paris. ISBN 2-87772-337-2
- Goeg., XI, 8, 2, Von Gutschmid.
- Seleucid-Parthian Studies, 1930, p 11; The Greeks in Bactria & India, 1938, p 292, William Woodthorpe Tarn
- Trade and Trade Routes in Ancient India, 1977, p 94, Moti Chandra; Geographical and Economic Studies in the Mahābhārata: Upāyana Parva, 1945, p 17, Moti Chandra.
- Proceedings of the British Academy, 1930, p 113, British Academy, Balasundara Gupta; Literary History of Ancient India in Relation to Its Racial and Linguistic Affiliations, 1953, p 148, Chandra Chakraberty – Sanskrit literature.
- "The Yüeh-chih and their migrations." K. Enoki, G. A. Koshelenko and Z. Haidary. In: History of civilizations of Central Asia, Volume II. The development of sedentary and nomadic civilizations: 700 B.C. to A.D. 250", p. 173. Harmatta, János, ed., 1994. Paris: UNESCO Publishing.
- "The Tokharians and Buddhism", p. 3. Xu Wenkan, In: Studies in Central and East Asian Religions 9, pp. 1–17 (1996). Downloaded on 14 June 2003, from: 
- A Study of Saka History, pp. 140–141. Taishan Yu. Sino-Platonic Papers No. 80. July, 1998. Dept. of Asian and Middle Eastern Studies, University of Pennsylvania.
- Taishan Yu. A Study of Saka History, pp. 12, 15, 24, 140. (1998) Sino-Platonic Papers. University of Pennsylvania.
- Taishan Yu. A Study of Saka History, pp. 21 and 38, n. 13 (1998) Sino-Platonic Papers. University of Pennsylvania.
- J. P. Mallory and Victor H.Mair. (2000) The Tarim Mummies, p. 92. Thames & Hudson Ltd., New York and London. ISBN 0-500-05101-1.
- J. P. Mallory and Victor H.Mair. (2000) The Tarim Mummies, pp.91–92. Thames & Hudson Ltd., New York and London. ISBN 0-500-05101-1.
- Goeg., XI, 8, 2, Alfred von Gutschmid.
- W. W. Tarn. The Greeks in Bactria and India. 2nd edition. (1951), pp. 284, 286, 533. Cambridge.
- Taishan Yu. A Study of Saka History, p. 40, n. 30. (1998) Sino-Platonic Papers. University of Pennsylvania.
- J. Markwart. Ērānšahr. (1901), p. 206. Referred to in: Taishan Yu. A Study of Saka History, p. 38, n. 17. (1998) Sino-Platonic Papers. University of Pennsylvania.
- Rannie tyurki. Ocerki istorii i ideologii. (2002) Yury Aleksey Zuev. Daik-Press, Almaty, Kazakhstan. In Russian. English title: Early Turks: Essays of History and Ideology. Draft translation by Norm Kisamov, ), p. 10. See also, pp. 21, 23, 29–30, 33–34.
- A. K. Narain. The Indo-Greeks, p. 132. (1957). Oxford University Press.
- Kazim Abdullaev (2007). "Nomad Migrations in Central Asia." In: After Alexander: Central Asia before Islam. Proceedings of the British Academy – 133, Eds. Joe Cribb & Georgina Herrmann, p. 75. ISBN 978-0-19-726384-6.
- Rapin, Claude (2007). "Nomads and the Shaping of Central Asia." In: After Alexander: Central Asia before Islam. Proceedings of the British Academy – 133, Eds. Joe Cribb & Georgina Herrmann, pp. 59–60. ISBN 978-0-19-726384-6.