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Several computer-aided diagnosis (CAD) systems based on machine and deep learning models have emerged in-order to assist clinicians in timely diagnosis of malignant (cancerous) melanoma. The conventional approaches derive low-level, handcrafted features from dermoscopy images of skin lesion. Novel deep learning-based neural networks are developed which aim to extract more generic and deep features for model training. Moreover, dermoscopy and clinical images play a central role in accurate detection of cancerous melanoma. This review paper is organized in five steps: First, we explain image analysis techniques of lesion images. Second, we highlight the challenges in identifying a lesion as melanoma or benign. Third, we provide an overview of publicly available skin lesion datasets. Fourth, we review the performance of various machine and deep learning-based melanoma diagnosis frameworks. This study exhibits that deep learning-based models and their ensemble outperform conventional machine learning approaches in respect of reliability, accuracy and sensitivity. 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