Here's your 10-point cheatsheet to this big story:
The election is being held amid tough campaigning by Prime Minister Narendra Modi, who is the de-facto face of the NDA, even though Chief Minister Nitish Kumar is the official candidate. PM Modi has so far held 10 rallies in the state, there are two more to go. The BJP has made the switch believing that PM Modi's popularity, unlike that of Nitish Kumar, is intact.
Tejashwi Yadav, 31, faces the BJP's Satish Kumar in Vaishali's Raghopur, where he trounced him in 2015 and wrested the seat. The BJP leader, who beat his mother Rabri Devi from the same seat in 2010, has been given a second chance by his party.
Asked if PM Modi's campaign is making him nervous, Tejashwi Yadav told NDTV, "I have nothing to lose. I am happy I fought on good issues. I have age on my side. I will try hard, work more".
Tejashwi Yadav's elder brother Tej Pratap Yadav has shifted base from Mahua in Vaishali and is trying his luck from Hasanpur in Samastipur district. The shift was linked to his marital discord with Aishwarya Rai. His estranged wife and father-in-law Chandrika Rai have been campaigning against the RJD candidate from the nearby Parsa seat.
The last few days have seen sharp jibes during campaigning. While PM Modi tagged Tejashwi Yadav as "Jungle Raj ka Yuvraj", Nitish Kumar mocked him and LJP's Chirag Paswan as a "cricketer and actor using him for publicity". Tejashwi Yadav has almost daily targeted the Chief Minister as "mentally and physically tired". Mr Kumar was also called "General Dyer" by the opposition over the police action during the Durga idol immersion in Munger, in which one person was shot dead.
All four assembly segments in state capital Patna held by the BJP -- Patna Sahib, Kumhrar, Bankipur and Digha -- are voting in the second phase. Seven constituencies of Nalanda district, to which Chief Minister Nitish Kumar belongs, are also up for election.
The biggest chunk of assembly seats where by-polls are being held is in Madhya Pradesh - 28. Two separate incidents of firing were reported in the Morena Sumawali seat in Madhya Pradesh, affecting the polling process briefly. Four people have been arrested. Eight Assembly seats are up for polls in Gujarat, seven in Uttar Pradesh, two each in Odisha, Nagaland, Karnataka, and Jharkhand; and one seat each in Chhattisgarh, Telangana, and Haryana.
The election will be crucial for Jyotiraditya Scindia, whose 22 loyalist followed him out of the Congress and into the BJP in March, pulling down the Kamal Nath government in Madhya Pradesh. Campaigning opposite Mr Scindia is his friend and former party colleague Sachin Pilot, whose brief rebellion against Rajasthan Chief Minister Ashok Gehlot -- coming soon after Mr Scindia's walkout -- had given Congress the jitters.
By-elections are being held on seven assembly seats in Uttar Pradesh. In 2017, the BJP won six of these. The most-watched will be Unnao's Bangarmau seat, where BJP's Kuldeep Sengar was disqualified after his conviction in a rape case.
The last phase of elections will be held on November 7, after which the votes will be counted on November 10.