The increase of fake news in social media, especially about Covid-19, poses a real threat to the mental and physical health of people. It is an important task to detect such news and to stop it spreading. In this article we describe the main approaches for fake news about Covid-19 detection, including Classical Machine Learning models, models based on Neural Networks and models, which were created based on the other approaches and preprocessing steps. We analyze the results of the challenge “Constraint@AAAI2021 - COVID19 Fake News Detection”, the main goal of which was the binary classification of news collected from social media for fake and real news. We analyze the best approaches which were proposed by researchers during the challenge. Also, we describe datasets of fake news related to Covid-19, which could be useful for the detection and classification of such news.