Over the years, Machine Learning models have been successfully employed on neuroimaging data for accurately predicting brain age. Deviations from the healthy brain aging pattern are associated with the accelerated brain aging and brain abnormalities. Hence, efficient and accurate diagnosis techniques are required to elicit accurate brain age estimations. Several contributions have been reported in the past for this purpose, resorting to different data-driven modeling methods. Recently, deep neural networks (also referred to as deep learning) have become prevalent in manifold neuroimaging studies, including brain age estimation. In this review, we offer a comprehensive analysis of the literature related to the adoption of deep learning for brain age estimation with neuroimaging data. We detail and analyze different deep learning architectures used for this application, pausing at research works published to date quantitatively exploring their application. We also examine different brain age estimation frameworks, comparatively exposing their advantages and weaknesses. Finally, the review concludes with an outlook towards future directions that should be followed by prospective studies. The ultimate goal of this paper is to establish a common and informed reference for newcomers and experienced researchers willing to approach brain age estimation by using deep learning models.