Propensity of skin diseases to manifest in a variety of forms, lack and maldistribution of qualified dermatologists, and exigency of timely and accurate diagnosis call for automated Computer-Aided Diagnosis (CAD). This study aims at extending previous works on CAD for dermatology by exploring the potential of Deep Learning to classify hundreds of skin diseases, improving classification performance, and utilizing disease taxonomy. We trained state-of-the-art Deep Neural Networks on two of the largest publicly available skin image datasets, namely DermNet and ISIC Archive, and also leveraged disease taxonomy, where available, to improve classification performance of these models. On DermNet we establish new state-of-the-art with 80% accuracy and 98% Area Under the Curve (AUC) for classification of 23 diseases. We also set precedence for classifying all 622 unique sub-classes in this dataset and achieved 67% accuracy and 98% AUC. On ISIC Archive we classified all 7 diseases with 93% average accuracy and 99% AUC. This study shows that Deep Learning has great potential to classify a vast array of skin diseases with near-human accuracy and far better reproducibility. It can have a promising role in practical real-time skin disease diagnosis by assisting physicians in large-scale screening using clinical or dermoscopic images.