I don't have my books in front of me, but here is a short answer that may be illuminating. The first Persian Empire, known as the Achaemenid Empire was founded in 550 by Cyrus the Great. It was later famously defeated by a Macedonian Army lead by Alexander the Great. After Alexander's death, his successors went to war, and divided the empire between themselves. One of his former generals, Seleucus Nicator found himself as ruler of the former Persian territories save Egypt and Anatolia.

This Hellenic Seleucid empire lasted only one hundred years, when the last Hellenic king was overthrown by a Parthian ruler of the Arsacid dynasty. Like Seleucus and his heirs, the Parthians were not ethnically Persian, but adapted their culture to become closer to the people they ruled.

The Parthian Empire ruled greater Persia for four hundred years, until the last Parthian king was overthrown by a Persian Satrap named Ardashir. This new ethnically Persian family, called the Sasaniads, worked to return Persia to a state of greatness, and during this reign of Ardashir's successor Shapur, Persia enjoys two great victories against the Roman Empire even capturing the Emperor Valerian in battle. The Sasaniads also made certain reforms to the state religion Zoroastrianism in order to solidify their control over an ethnically diverse population. The Sasaniad empire lasted for an additional 400 years, ruling as a ethnically Persian Empire that clashed frequently with the Roman Empire to the determent of both.

However In 632 a new world superpower arose from the depths of the Arabian desert, and swept down upon the empires of Rome and Persia achieving victories of frightening proportions. The newly created Rashidun Caliphate, led by the popular magnate Abu Bakr, and with an army commanded by the legendary Khalid ibn al-Walid, the Muslim armies invaded both empires. Over the span of twenty years the forces of the Caliphate conquered Palestine, Syria, and Egypt from the Roman Empire, while in the west the entire Sasanian was defeated, and replaced with Islamic institutions. One of the big changes in Persia after the Islamic conquest was the marginalization of the formerly powerful Zoroastrian priests, who lost their positions of royal patronage. As such, conversion to Islam for the elite was required to serve in the Caliphs government, and though the local peasants were not forcibly converted during this period, both the Zoroastrian religion, and the historical Persian identity as an empire ruled by the King of Kings had been broken.

In Conclusion the Persian Empire was defeated twice, once at the end of the Achaemenid when they were conquered by Alexander, and a second time after the Sasanian Empire fell to the Caliphate.

Youll need to specify the question first.

There have been various Persian dynasties throughout history, so it is not clear what time period you are referring to.