From The Myth
According to Swayambhu Purana, the entire valley was once filled with an enormous lake, out of which grew a lotus. The valley came to be known as Swayambhu, meaning “Self-Created.” The name comes from an eternal self-existent flame over which a supa was later built.
Swayambhunath is also known as the Monkey Temple as there are holy monkeys living in the north-west parts of the temple. They are holy because Manjushri, the bodhisattva of wisdom and learning was raising the hill which the stupa stands on. He was supposed to leave his hair short but he made it grow long and head lice grew. It is said that the head lice transformed into these monkeys.
Manjusri had a vision of the Lotus at Swayambhu and traveled there to worship it. Seeing that the valley can be a good settlement and to make the site more accessible to human pilgrims, he cut a gorge at Chovar. The water drained out of the lake, leaving the valley in which Kathmandu now lies. The Lotus was transformed into a hill and the flower became the stupa.
From The History
Swayambhunath is among the oldest religious sites in Nepal. According to the Gopālarājavaṃśāvalī , it was founded by the great-grandfather of King Manadeva (464-505 CE), King Vṛsadeva, about the beginning of the 5th century CE. This seems to be confirmed by a damaged stone inscription found at the site, which indicates that King Manadeva ordered work done in 640 CE.
However, Emperor Ashoka is said to have visited the site in the third century BCE and built a temple on the hill which was later destroyed.
Although the site is considered Buddhist, the place is revered by both Buddhists and Hindus. Numerous Hindu monarch followers are known to have paid their homage to the temple, including Pratap Malla, the powerful king of Kathmandu, who is responsible for the construction of the eastern stairway in the 17th century.
The stupa was completely renovated in May 2010, its first major renovation since 1921 and its 15th in the nearly 1,500 years since it was built. The dome was re-gilded using 20 kg of gold. The renovation was funded by the Tibetan Nyingma Meditation Center of California, and began in June 2008.
At around 5 a.m. on 14 February 2011, Pratapur Temple in the Swayambhu Monument Zone suffered damage from a lightning strike during a sudden thunderstorm.
The Swayambunath complex suffered damage in the April 2015 Nepal earthquake.
The stupa consists of a dome at the base, above which is a cubical structure painted with eyes of Buddha looking in all four directions. There are pentagonal Toran present above each of the four sides with statues engraved in them. Behind and above the Torana there are thirteen tiers. Above all the tiers there is a small space above which the Gajur is present. The stupa has many artifacts inside it.
Some important monuments to be seen in these area .
The Sleeping Buddha
The Statue of the Lord Buddha at the western part of the Swayambunath .
At the east side of the stupa situated huge gold plated Vajra ‘thunderbolt’.
And Many more ….
Source : https://en.wikipedia.org/wiki/Swayambhunath