Here are the 10 latest developments in this story:
The BJP is leading in 65 of 103 wards in MCD North, 67 of 103 in MCD South and 49 of 63 in MCD East, well over halfway in all three. Overall, out of a total 272 wards, the BJP is leading in 181, AAP in 46 and the Congress in 29 wards.
In each municipality the BJP has bettered last time's performance, up seven wards in MCD North, 12 wards in MCD South and 14 in MCD East. The MCD was split into three municipalities in 2012.
"I thank Delhi's voters for the BJP's win," said BJP chief Amit Shah crediting Prime Minister Narendra Modi's governance for the victory. "The people of Delhi have rejected negative politics, the politics of excuses," Mr Shah said.
"This is an EVM wave not a Modi wave," said AAP's Delhi minister Gopal Rai, alleging that EVMs or electronic voting machines were rigged to help the BJP win the MCD elections. Delhi's Deputy Chief Minister Manish Sisodia said, "BJP's win was impossible without EVM tampering."
The BJP has called AAP a poor loser. "Kejriwal is a drama queen," said the BJP's Shazia Ilmi, a former member of AAP. "Kejriwal thought he would become PM after becoming CM of Delhi. He is losing the election just because of his aggression," said the BJP's Sambit Patra.
For AAP, today's loss means a political wipe-out. Chief Minister Kejriwal's party, which had swept assembly elections in Delhi two years ago, is already reeling from humiliating losses in the Punjab and Goa assembly elections last month.
The Congress had hoped today's results would herald a political revival in Delhi for the party, which did not win a single seat in the Delhi assembly elections in 2015 or a parliament seat the year before.
Ajay Maken said he was quitting as the Congress' Delhi chief and would not hold any party post for a year. He said the Congress was the day's big gainer on vote share, but he had expected to do better. Just ahead of the MCD polls some top Delhi Congress leaders quit the party and joined the BJP as a rebellion bubbled against Mr Maken.
The MCD elections is an important addition to the BJP's list of victories this year in state and local polls across the country. The party has dominated the MCD for a decade and conducted a big campaign to retain control, still smarting at being reduced to just three seats in the Delhi assembly in 2015, as AAP swept 67 of the 70 seats. In the 2014 national election, the BJP had won all seven of Delhi's parliament seats.
In a strategic move the BJP had sought to eliminate all chances of an anti-incumbency sentiment by fielding new faces in 267 wards of the total 272 wards.