COVID-19 is a novel virus that causes infection in both the upper respiratory tract and the lungs. The numbers of cases and deaths have increased on a daily basis on the scale of a global pandemic. Chest X-ray images have proven useful for monitoring various lung diseases and have recently been used to monitor the COVID-19 disease. In this paper, deep-learning-based approaches, namely deep feature extraction, fine-tuning of pretrained convolutional neural networks (CNN), and end-to-end training of a developed CNN model, have been used in order to classify COVID-19 and normal (healthy) chest X-ray images. For deep feature extraction, pretrained deep CNN models (ResNet18, ResNet50, ResNet101, VGG16, and VGG19) were used. For classification of the deep features, the Support Vector Machines (SVM) classifier was used with various kernel functions, namely Linear, Quadratic, Cubic, and Gaussian. The aforementioned pretrained deep CNN models were also used for the fine-tuning procedure. A new CNN model is proposed in this study with end-to-end training. A dataset containing 180 COVID-19 and 200 normal (healthy) chest X-ray images was used in the study's experimentation. Classification accuracy was used as the performance measurement of the study. The experimental works reveal that deep learning shows potential in the detection of COVID-19 based on chest X-ray images. The deep features extracted from the ResNet50 model and SVM classifier with the Linear kernel function produced a 94.7% accuracy score, which was the highest among all the obtained results. The achievement of the fine-tuned ResNet50 model was found to be 92.6%, whilst end-to-end training of the developed CNN model produced a 91.6% result. Various local texture descriptors and SVM classifications were also used for performance comparison with alternative deep approaches; the results of which showed the deep approaches to be quite efficient when compared to the local texture descriptors in the detection of COVID-19 based on chest X-ray images.