Shani Jayanti, or Sani Jayanthi, is believed to be the day when Lord Shani, one of the Navagrahas made its appearance on Earth. In 2016, the date of Shani Jayanti is June 5, Sunday. Shani is feared by those people who believe and follow astrology, as Shanaishchara has the capacity to create malefic influence on the horoscope and thus in their life.
On Shani Jayanti day, people perform special pujas at Navagraha Temples or in Shani Temples. Those Hindus having Saturn cycle or Shani cycle or Sade Sathi in their horoscope perform special pujas like Shani Yagya or Homa. Some devotees observe a fast (fasting or upvaas) on the day and the rules observed are similar to Shanivar Vrat or Saturday Fasting.
You can find more details about Shanaishchara Jayanthi in this article.
To overcome the fear of Shan all you need to do is offer prayers to Hanumanji.
Offerings to be made in Shani Temple - Black cloth, black til (sesame), and til oil.
It must be noted that there are numerous crooks in the guise of astrologers, pandits and pujaris who take undue advantage of innocent people on such occasions. They put the blame for all misfortune on Shani and try to conduct expensive pujas etc. Beware of such crooks.
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