The Greeks and Persians fought in many battles between 499 BCE and 449 BCE. However two great Greek victories stand out. The first, a sea battle in 480 BCE was fought at Salamis. The second was a land battle at Plataea the following year. These two battles defeated the Persian invasion of mainland Greece.
It was not the Greeks as such. The Persian empire included the Greek cities in Asia Minor - mostly Ionian Greek colonies of the Ionian mother cities of Athens and those in Euboia including Eretria. These mother cities tended to support their colonies on the far side of the Aegean Sea when they revolted against Persian rule, and when Miletus revolted, Athenian and Eretrian forces were sent to support it, in the course of which they provocatively raided and burnt the Persian provincial capital of Sardis. Exceedingly angry, Persian king Darius I sent a punitive amphibious expedition against Athens and Eretria (not the mainland Greeks in general). The latter was captured, but Athens was successful in defeating the Persian force at the battle of Marathon. They were successful because the Persians split their forces. They intended to install a puppet in Athens - the expelled ex-tyrant of Athens Hippias, who arranged for his faction in the city to open the gates while the Athenian army was opposing the landing on the plain of Marathon. The plan was for the Persian cavalry to re-embark and row around the Sunion peninsula to land, gallop up through the opened gates and capture the city, leaving the Athenian army locked out and attacked from the rear by the Persian land force. The 9,000 Athenians at Marathon with 1,000 Plataian allies hung around the hills, fearful of engaging the Persians with their cavalry support. On the tenth day, they noted the horses being loaded on the ships and seized the opportunity to run down and engage the unprotected Persian infantry with their own superior armoured warriors. They defeated the infantry, but the ships got away. Guessing what was on, the Athenian warriors ran back to Athens the 26 miles over the hills just in time to form up in front of the city as the Persian cavalry disembarked. The Persians reembarked, and with their land force badly damaged, went home. PS: This run over the hills was the real first Marathon - 26 miles with armour and weapons, having already fought a battle in the morning. Today's marathon runners get it pretty easy. The story of Pheidippides being the first marathon runner is incorrect - however he did it pretty tough. He was sent to Sparta to summon help, 120 mile round trip - he understandably came home hallucinating and died. The Spartans were celebrating a religious festival at the time, but as soon as it was over hit the road and got the 60 miles to Athens in a day, too late for the battle, however they marched on to view the battlefield, congratulated the Athenians, then marched back home. PPS: Darius was not at all pleased, as all this had just made the Greeks more cocky. He therefore planned to capture all of mainland Greece and install puppet regimes to stop this interference in his Asian empire, by establishing an ethnic frontier there. This plan was delayed by his having to put down a revolt in Egypt, and he then died before putting it into effect, however his son Xerxes took on the project, invading in 480 BCE. But that is another story.
It was the Athenians with their ally Plataia.
They defeated the weaker Persian infantry while their cavalry was absent.
The Greeks lost regularly in the war between 499 and 493 BCE.
The punitive raid against Eretria and Athens for intervening in those failed Greek uprisings within the Persian Empire was successful against Eretria but failed against Athens which defeated the Persian army at Marathon when they caught it with its protective cavalry absent.
In a sea battle at Salamis 480 BCE, a land battle at Plataea and a sea-land battle at Mycale in 479 BCE. Thereafter there were other battles, finally at Cyprus in 450 BCE, after which peace was made in 449 BCE.
The Battle of Marathon . This was a notable victory for the Greeks over the Persians .
Salamis 480 BCE.
At Granicus, Issus and Gaugamela.
On the plain of Marathon.
The strong army that Alexander defeated was the Persian Army
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There were several. The most decisive was Gaugamela.
when he defeat the king darius 3
The attack was no surprise - the two armies were confronting each other. Alexander wanted to defeat the Persian army to gain control of the Persian empire.
Gaugamela.
Granicus, then Isus, then Gaugamela.
The battle of Marathon 490 BCE.
Granicus
Alexander the Great decisively defeated the Persians led by Darius the Third at Gaugamela (located in Iraq) October 1, 331 BC .
Alexander the Great defeated Darius, king of Persia in 331 BCE. This was at the battle Gaugamela, he did march through Bactria in 329 BCE, but he had already defeated the Persians.
I'm almost positive that he defeated the Persians at the Battle of Gaugamela.