The most popular view of the history of Deepavali (or Diwali) is taken from Ramayana, the great hindu epic of India.
The epic says that after a long and tough fight, Lord Rama was finally able to kill the asura king Ravana, and bring back his wife, mother Sita. Ravana was slain on Dussera, which also celebrated all over India, when effigies with fireworks of Ravana are burnt, signaling triup of good over evil.

It was also the end of his exile (vanabas) and time to return to his kingdom.
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The day Rama came back to his kingdom, with his brother Laxman, and wife Sita, his whole kingdom was ecstatic with boundless with joy. They loved their king and the fact the king was returning resulted is celebrations everywhere. The whole kingdom was lit up with small diyas (oil lamps). Every house, every balcony, every street, and every place were lit up with numerous small lamps, welcoming Lord Rama. This Pleased the returning trio immensely.

The tradition is carried on till date, and the whole if India is lit up with small lights, celebrating the day of Lord Rama's return, and begining of prosperity.