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In the past two years, medical researchers and data scientists worldwide have focused their efforts on containing the pandemic of coronavirus disease 2019 (COVID-19). Deep learning models have been proven to be capable of efficient medical diagnosis and prognosis in cancer, common lung diseases, and COVID-19. On the other hand, artificial neural networks have demonstrated their potential in pattern recognition and classification in various domains, including healthcare. This literature review aims to report the state of research on developing neural network models to diagnose COVID-19 from cough sounds to create a cost-efficient and accessible testing tool in the fight against the pandemic. A total of 35 papers were included in this review following a screening of the 161 outputs of the literature search. We extracted information from articles on data resources, model structures, and evaluation metrics and then explored the scope of experimental studies and methodologies and analyzed their outcomes and limitations. We found that cough is a biomarker, and its associated information can determine an individual's health status. Convolutional neural networks were predominantly used, suggesting they are particularly suitable for feature extraction and classification. The reported accuracy values ranged from 73.1% to 98.5%. Moreover, the dataset sizes ranged from 16 to over 30,000 cough audio samples. Although deep learning is a promising prospect in identifying COVID-19, we identified a gap in the literature on research conducted over large and diversified data sets. [ABSTRACT FROM AUTHOR]