SARS-CoV-2 is a novel virus that has been affecting the global population by spreading rapidly and causing severe complications, which require prompt and elaborate emergency treatment. Automatic tools to diagnose COVID-19 could potentially be an important and useful aid. Radiologists and clinicians could potentially rely on interpretable AI technologies to address the diagnosis and monitoring of COVID-19 patients. This paper aims to provide a comprehensive analysis of the state-of-the-art deep learning techniques for COVID-19 classification. The previous studies are methodically evaluated, and a summary of the proposed convolutional neural network (CNN)-based classification approaches is presented. The reviewed papers have presented a variety of CNN models and architectures that were developed to provide an accurate and quick automatic tool to diagnose the COVID-19 virus based on presented CT scan or X-ray images. In this systematic review, we focused on the critical components of the deep learning approach, such as network architecture, model complexity, parameter optimization, explainability, and dataset/code availability. The literature search yielded a large number of studies over the past period of the virus spread, and we summarized their past efforts. State-of-the-art CNN architectures, with their strengths and weaknesses, are discussed with respect to diverse technical and clinical evaluation metrics to safely implement current AI studies in medical practice.