The Ionian Revolt of the Greek city-states within the Persian Empire 499 to 493 BCE, which was put down by Persia.
Darius I led the Persians in the First Persian War. Xerxes I led them in the Second.
A few reasons, First, the Greeks had 7,000 men and the Persians had 15,000 Second, the Greeks fought the Persian Immortals which were histories most elite and powerful warriors. Third, the Persians found a route to outflank the Greeks.
The Battle of Marathon was a turning point during the first Persian invasion of Greece. The Persians vastly outnumbered the Greeks on the battlefield, but the Greeks were able to defeat them. The Greeks charged the Persian troops with a thin weaker line, while the Greeks' left and right flanks consisted of stronger troops who quickly surrounded the Persian troops and attacked them on both sides. It was a crushing defeat for the Persians, and the battle convinced the Greeks that while the Persian Empire had vast armies and archers, it was possible to defeat them.
They first captured the Mediterranean sea coast to eliminate the Persian navy, They also slaughtered Greek mercenaries as a warning for Greeks not to work for the Persians, denying the Persians reinforcement of armoured infantry. They then defeated the Persian army at Gaugamela, using armoured soldier who out-performed the unarmoured Persian infantry.
There were several - the most notable were Salamis (in the Saronic Gulf between the island of Salamis and Athens in 480; Mykale in Asia Minor 479 BCE and Eurymedon in Asia Minor 466 BCE.
They adopted a superior strategy of first defeating the Persian navy, and cutting off their sea supply line. The Persians had to send half their army home for the winter, and the Greeks united the city-state armies to defeat the depleted Persian army.
The combined Greek city-states of the eastern Mediterranean.
Darius I.
Alexander the Great first captured its Mediterranean ports so that it could no longer pose a naval threat. He massacred the Greek mercenaries in Persian service after the battle of the Granicus so that Greeks would no longer provide the armoured infantry the Persians needed, and he captured the Persian treasury after the battle of Issus. These measures stripprd the Persians of their underlying strengths.
Marathon - it was the Athenians.
Salamis.
Greeks could have attacked before the Persians but a lot of the evidence points to the Persians attacking first, the Persians were most likely to have fought when they got off their boats and when they encountered in the valley, but, the Greeks were to have probably ran away into the valley than to have been chased.