What team won the last Persian war and how?

The First Persian War ended with Persia abandoning its punitive expedition against Athens after the Battle of Marathon. The Second Persian War ended with the Persian Navy withdrawing and abandoning the army to its fate, leading to the Greek military victory at Platæa and the Greek naval victory at Mycale practically simultaneously, though many miles apart. Finally, Alexander the Great conquered the Persian Empire defeating Darius III and claiming all its territory for himself. The reasons are many, varied, and complicated and fill several volumes of histories.