The full acceptance of Deep Learning (DL) models in the clinical field is rather low with respect to the quantity of high-performing solutions reported in the literature. End users are particularly reluctant to rely on the opaque predictions of DL models. Uncertainty quantification methods have been proposed in the literature as a potential solution, to reduce the black-box effect of DL models and increase the interpretability and the acceptability of the result by the final user. In this review, we propose an overview of the existing methods to quantify uncertainty associated with DL predictions. We focus on applications to medical image analysis, which present specific challenges due to the high dimensionality of images and their variable quality, as well as constraints associated with real-world clinical routine. Moreover, we discuss the concept of structural uncertainty, a corpus of methods to facilitate the alignment of segmentation uncertainty estimates with clinical attention. We then discuss the evaluation protocols to validate the relevance of uncertainty estimates. Finally, we highlight the open challenges for uncertainty quantification in the medical field.