The north-eastern state of Assam also goes to the polls on Saturday in the first of three phases, while Tamil Nadu, Kerala and Puducherry vote on 6 April. Results from all are due on 2 May.
The campaign has been marked by violence with the BJP claiming that more than 100 of its workers have been killed over the last two years, with the Trinamool making similar claims
The election in West Bengal is the most important with the Hindu nationalist BJP pushing hard to win power in the Bengali-speaking region for the first time.
But the party faces a tough opponent in incumbent chief minister Mamata Banerjee, 66, a firebrand whose Trinamool Congress party ended three decades of rule in 2011.
The campaign has been marked by violence with the BJP claiming that more than 100 of its workers have been killed over the last two years, with the Trinamool making similar claims.
Activists from both parties have been shot or hacked to death, their bodies sometimes hung from trees.
Crude bombs, available on the black market for as little as 100 rupees ($1.40), have also been used to kill, maim or intimidate voters.
The mutilated body of one BJP activist, Sukhdev Pramanik, was found face-down in a pond in the village of Chandpara in December.
Sumita, the mother of Shoubhik Dolai, a Trinamool activist killed last month, told AFP that “they pumped bullets into him... I got to know when I saw the news on TV.”
Arati Jerath, a political analyst, said Banerjee has been at the forefront of trying to form an “anti-Modi opposition front”.
In Assam, the BJP leads an alliance and is hopeful of retaining power against a strong coalition of Congress and smaller regional parties.
The state is a polarised polity along ethnic and religious lines, with illegal migration from Bangladesh one of the biggest campaign issues.
A “citizenship list” in Assam state left off almost two million people who were unable to prove they were Indian, many of them Muslims, a process many fear the BJP wants to roll out nationwide.