Here are 10 facts in this big story:
Mr Advani, Mr Joshi and a dozen others including Vinay Katiyar and Kalyan Singh - the Governor of Rajasthan - were freed by a court in Raebareli of conspiracy charges in the razing of the 16th century Babri mosque in Ayodhya in December 1992 by Hindu activists who believed it was built on the birthplace of Lord Ram.
In 2010, the Allahabad High Court confirmed the lower court's decision to drop charges against the BJP veterans. The CBI has challenged the high court order.
On March 6, the Supreme Court said, "We will not accept the discharge of Advani and others on technical grounds. We will allow you (CBI) to file a supplementary charge-sheet against 13 persons by including the conspiracy charges. We will ask the trial court to conduct a joint trial."
Mr Advani's lawyer had told the Supreme Court that if conspiracy charges are added, then 183 witnesses who had testified in a lower court would have to be called again.
Of the two cases in the Babri demolition, one is against Mr Advani and the other BJP leaders who were on the dais at a raised platform called "Ram Katha Kunj" around 200 metres from the mosque site, where many provocative speeches were allegedly made.
The other case is against activists who called themselves karsevaks, accused of pulling down the three domes of the Babri mosque in an act that changed the face of Indian politics and caused a deep rift between two communities. The case is pending in a Lucknow trial court.
Mr Advani and Mr Joshi, who emerged as the BJP's most prominent faces along with former prime minister Atal Bihari Vajpayee at the height of the temple-mosque dispute, have been away from active politics after being assigned only a mentorship role in 2014, after Prime Minister Narendra Modi came to power.
The Babri Masjid was built in Ayodhya in 1528. Hindu groups believe it was built after demolishing a temple. After it was razed, a makeshift temple came up at the site.
The Allahabad High Court judgment in 2010 had said Lord Ram was born under the central dome of the makeshift temple in Ayodhya and Hindus have the right to worship there. That order was put on hold by the Supreme Court.
Yesterday, the Supreme Court suggested an out of court settlement to resolve what it called a "sensitive and sentimental issue". The Chief Justice of India, JS Khehar even said he is ready to mediate.