Ten more people were missing and the number of dead looked set to climb as large parts of the country were under water, officials said.
Rescue teams of soldiers, police and local people have been deployed to help those trapped on rooftops or on higher ground, officials added. TV pictures showed damaged roads and bridges.
Weather experts said more rain was expected overnight and the government has told residents to remain alert. The Koshi river in the south of the country, a main waterway, was above the danger level, authorities warned.
Monsoon rains from June to September are important for farm-dependent Nepal, but they also cause havoc each year.
'It has left extensive damage,' Dipak Kafle, a spokesman for Nepal's Home Ministry, told Reuters. He said there was no estimate yet for property losses.
'We have more than 30 people confirmed dead. But we are more focused on rescuing those who are alive than collecting details about the dead,' he said, indicating the death toll could go up.
In the eastern city of Biratnagar, an airport runway was under 2 feet of water, TV footage showed, leading to a suspension of flights. Highways were either damaged or inundated, disrupting transport, police said.
Local officials have opened temporary shelters for displaced people in schools or public buildings.