Twenty-one deaths occurred in Mirik alone, where another 13 people were also reported missing, officials in West Bengal's Disaster Management Department said.
The remaining 17 died in Kalimpong-1 and 2, Lava, Sukhia block and Gorubathan, they said.
The landslides in 25 places in Darjeeling, Kalimpong and Kurseong sub-divisions also caused extensive damage to NH-10 and NH-55, cutting off road link to the region.
Sashastra Seema Bal (SSB) personnel were carrying out rescue operations and the Army's help has been sought.
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Border Roads Organisation (BRO) men were engaged in clearing the blocked roads.
Chief Minister Mamata Banerjee, who is now in Murshidabad district, is on way to the affected places.
"I am monitoring the situation closely and will go there after the meeting in Murshidabad," she tweeted.
She said that the home secretary was already in North Bengal.
Prime Minister Narendra Modi directed Union Minister Kiren Rijiju to visit landslide-hit Darjeeling to take stock of the situation.
NH 10 bore the brunt with the road link to Kalimpong, Lava, Lolegaon and Garubathan snapped. NH 55 connecting Siliguri, Matigara with Darjeeling was also damaged mostly in Mirik and Rohini areas. A bridge on NH 55 has been washed away at Nimbujhora, the officials said.
Landslides occurred at several places between Mongpong and Sevoke Kalibari, leaving people stranded on NH 31 which links these places.