COVID-19 is an acute severe respiratory disease caused by a novel coronavirus SARS-CoV-2. After its first appearance in Wuhan (China), it spread rapidly across the world and became a pandemic. It had a devastating effect on everyday life, public health, and the world economy. The use of advanced artificial intelligence (AI) techniques combined with radiological imaging can be helpful in speeding-up the detection of this disease. In this study, we propose the development of recent deep learning models for automatic COVID-19 detection using computed tomography (CT) images. The proposed models are fine-tuned and optimized to provide accurate results for multiclass classification of COVID-19 vs. Community Acquired Pneumonia (CAP) vs. Normal cases. Tests were conducted both at the image and patient-level and show that the proposed algorithms achieve very high scores. In addition, an explainability algorithm was developed to help visualize the symptoms of the disease detected by the best performing deep model.