This paper explores how well deep learning models trained on chest CT images can diagnose COVID-19 infected people in a fast and automated process. To this end, we adopt advanced deep network architectures and propose a transfer learning strategy using custom-sized input tailored for each deep architecture to achieve the best performance. We conduct extensive sets of experiments on two CT image datasets, namely the SARS-CoV-2 CT-scan and the COVID19-CT. The obtained results show superior performances for our models compared with previous studies, where our best models achieve average accuracy, precision, sensitivity, specificity and F1 score of 99.4%, 99.6%, 99.8%, 99.6% and 99.4% on the SARS-CoV-2 dataset; and 92.9%, 91.3%, 93.7%, 92.2% and 92.5% on the COVID19-CT dataset, respectively. Furthermore, we apply two visualization techniques to provide visual explanations for the models' predictions. The visualizations show well-separated clusters for CT images of COVID-19 from other lung diseases, and accurate localizations of the COVID-19 associated regions.