According to the authorities, around a hundred and fifty people lived at the site and dozens of them are still missing.
Many people at the landfill were "frantically looking for friends and family", Hope for Korah, a Canadian NGO that assists people living in the area near the dump, said on its Facebook page.
City officials say close to 300,000 tons of waste are collected each year from the capital, most of it dumped at the landfill.
The mayor of Addis Ababa noted that more than three dozen people were rescued and received medical treatment. Elderly women cried, and others stood anxiously waiting for news of loved ones.
She said that 28 people were injured, two seriously.
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Tebeju Asres, an eyewitness said that his family's house was buried under the rubbish.
Officials said about 150 people were present at the large landfill when the slide happened. The resumption of rubbish being dumped here was the probable cause of the landslide, local resident Assefa Teklemahimanot told AP. The government had also been building a factory to convert waste products at the landfill into electric energy, he said. Some 500 waste pickers are believed to visit the dump every day, sifting through the waste of the better off for anything of value.
Ethiopia has one of Africa's fastest-growing economies and is predicted by the World Bank to become a middle-income country by 2025.
Many of them are huddled in small groups talking in low tones, others crying and sobbing loudly.