Reading between the lines of the ancient historians, it is clear that, at least at the beginning, Zenobia thought that her military activities in the eastern part of the empire were "doing Rome's work" and that she certainly thought that Rome would not object to her "defense of the empire." She had been, after all, the wife of a client king, Odaenathus of Palmyra. It was Odaenathus who harried the retreating Persians after they had captured the "senior" Emperor Valerian at Rome's disastrous defeat at Carrhae (260 AD). At that point the junior co-emperor, Gallienus, confirmed Odaenathus as supreme commander in charge of defense of the eastern frontier, and Odaenathus then pursued the Persians into their own territory winning victories in the name of Rome. Gallienus heaped honors on Odaenathus and, at least in eastern monuments, Odaenathus designated himself "Senator of Rome" and "Consul".
He obviously was advertising his connections with Rome as a means of increasing his local support. Despite these efforts, there were plots against him, and he was assassinated in 267 AD along with his oldest son. Details of the assassination have always been murky, but Zenobia has sometimes been blamed, because her own young son by Odaenathus succeeded to the Palmyrine throne. Although the boy, Vabalathus, got the throne, Zenobia succeeded to the actual power as his regent.
Meanwhile, back in Rome, Aurelian restored order in the city and finished his walls. Then he reconquered Gaul. When he turned his attention eastward, he initially granted Zenobia's request that her son be named "Leader (Duke) of the Romans." In short order Zenobia's troops took control of Rome's fractious Syrian and Egyptian provinces, and most of Asia Minor, ostensibly in the name of Rome. According to contemporary accounts, she was an able strategist and personally led her own and allied native troops into battle. Throughout this period, coins were issued by mints in the east, under Zenobia's control, that showed Valabalathus as a vassal of Aurelian. There was a hint of Roman displeasure when the Roman Governor of Egypt tried unsuccessfully to confront Zenobia's local commander.
Aurelian became suspicious. He was being prodded into action by public opinion (mob and Senate) and his own ego: the "Palmyrene Revolt" was perhaps a serious threat to his sovereignty. In 271 AD Roman legions led by Aurelian moved against Zenobia's mini-empire.
The Romans initially were overconfident -- one legion moved too soon and was destroyed when it attacked perhaps 80,000 of Zenobia's forces. And there were other early Zenobia victories -- apparently also in battles started by the Romans. Eventually, however, Aurelian mustered a large enough force, and a decisive battle was fought east of Antioch. Aurelian's light cavalry lured Zenobia's more heavily armored cavalry into an overheated chase, and, when they were thoroughly baked, the Roman cavalry turned and, aided by Aurelian's infantry, cut the Palmyrine cavalry to shreds. Zenobia fled back through Antioch with the Roman armies in pursuit.
A similar tactic at a battle on the plains of Emesa took down another important Palmyrine heavy cavalry force. Zenobia again had to flee -- without her treasury, and this time all the way back to Palmyra.
Aurelian besieged that city, and eventually Zenobia slipped out with a small escort and fled on camelback toward the Euphrates, the Persian border. She was caught and captured by a Roman light cavalry unit at the river's edge. Palmyra surrendered when she was displayed in chains outside the city. Aurelian spared the population but took its wealth and leaders back for trial at Emesa. Most were executed, but Zenobia was brought back to Rome as the centerpiece of Aurelian's "Triumph" of 274 AD.
Zenobia, along with captives from Aurelian's other campaigns, was marched along Rome's triumphal route, from the Campus Martius, through the Circus Maximus, down to the Meta Sudans (a fountain near where the Arch of Constantine was later built), and then through the Forum Romanum. Her chains were said to have been of gold and so heavy that her attendants had to help her carry them. Later art work shows her semi-naked, but that says more about the perversity of the later artists than about the actual circumstances.
Clothed or not, she was apparently well treated after the triumph. By most accounts she lived out her life in wealth and comfort as the matron of a sumptuous villa in Tibur (modern Tivoli). Some accounts say she married a Roman senator, but that might just have been confusion caused by the fact that Odaenathus, her husband back in the "old country", had used that title.
Zenobia today: There are still Syrians who regard Zenobia as an archetype for Syrian aspirations. Just as the British have memorialized Buodicca, the Syrians have taken Zenobia as an exemplar, their earliest militant anti-colonialist. Some Syrian irredentists invoke her example in advocating recovery of sections of "Greater Syria" (which, they say, included Syria, Lebanon, Palestine, and parts of Turkey). On a more mundane level Palmyra is being promoted as a serious Syrian tourist destination, with emphasis on the accomplishments of Zenobia.
The best short modern account on the web is at http://www.roman-emperors.org/zenobia.htm.
An English paraphrase of Vopiscus's Aurelian's Conquest of Palmyra is at: http://www.fordham.edu/halsall/ancient/vopiscus-aurelian1.html, but Vopiscus didn't, of course, have the benefit of modern historical research, particularly from eastern sources.
Palmyra, Government of Syria tourism promotion site: http://www.syriatourism.org/Destinations/palmyra.htm