The landslides occurred on Thursday night and Friday morning in central, southern and western parts of the country after two days of rain, the country’s Disaster Management Centre (DMC) said.
Eleven bodies were recovered after a landslide in Bulathsinhala, 65km southeast of the capital, Colombo.
“Twenty deaths and 66 missing reported … as of 3pm (10:30 GMT),” the DMC said in a statement.
Heavy rain also flooded several parts of the southern Galle and Matara districts – both popular tourist destinations.
Mudslides in the Kalutara district, south of Colombo, buried houses and shops, local media reported.
The rains also inundated major roads in the region, causing disruption in transport services and hampering rescue boat operations, officials said.
The country’s navy deployed more than 100 sailors and 20 boats to rescue and help those affected.
Authorities warned people living in low-lying areas close to major rivers to evacuate immediately amid fears of rising water levels.
The heavy rains and floods also prompted the closure of all schools in the Sabaragamuwa province, about 90km east of Colombo.
Mudslides are common during the monsoon season in Sri Lanka, an Indian Ocean tropical island.
Much of the country has been heavily deforested for export crops such as tea and rubber, leaving the countryside exposed.
In May 2016, at least 127 people were killed in a landslide in the central Kegalle district.