Data protection issues stemming from the use of machine learning algorithms that are used in automated decision-making systems are discussed in this paper. More precisely, the main challenges in this area are presented, putting emphasis on how important it is to simultaneously ensure the accuracy of the algorithms as well as privacy and personal data protection for the individuals whose data are used for training the corresponding models. In this respect, we also discuss how specific well-known data protection attacks that can be mounted in processes based on such algorithms are associated with a lack of specific legal safeguards; to this end, the General Data Protection Regulation (GDPR) is used as the basis for our evaluation. In relation to these attacks, some important privacy-enhancing techniques in this field are also surveyed. Moreover, focusing explicitly on deep learning algorithms as a type of machine learning algorithm, we further elaborate on one such privacy-enhancing technique, namely, the application of differential privacy to the training dataset. In this respect, we present, through an extensive set of experiments, the main difficulties that occur if one needs to demonstrate that such a privacy-enhancing technique is, indeed, sufficient to mitigate all the risks for the fundamental rights of individuals. More precisely, although we manage—by the proper configuration of several algorithms’ parameters—to achieve accuracy at about 90% for specific privacy thresholds, it becomes evident that even these values for accuracy and privacy may be unacceptable if a deep learning algorithm is to be used for making decisions concerning individuals. The paper concludes with a discussion of the current challenges and future steps, both from a legal as well as from a technical perspective.