A lot of research has been conducted in recent years on stereo depth estimation techniques, taking the traditional approach to a new level such that it is in an appreciably good form for competing in the depth estimation market with other methods, despite its few demerits. Sufficient progress in accuracy and depth computation speed has manifested during the period. Over the years, stereo depth estimation has been provided with various training modes, such as supervised, self-supervised, and unsupervised, before deploying it for real-time performance. These modes are to be used depending on the application and/or the availability of datasets for training. Deep learning, on the other hand, has provided the stereo depth estimation methods with a new life to breathe in the form of enhanced accuracy and quality of images, attempting to successfully reduce the residual errors in stages in some of the methods. Furthermore, depth estimation from a single RGB image has been intricate since it is an ill-posed problem with a lack of geometric constraints and ambiguities. However, this monocular depth estimation has gained popularity in recent years due to the development in the field, with appreciable improvements in the accuracy of depth maps and optimization of computational time. The help is mostly due to the usage of CNNs (Convolutional Neural Networks) and other deep learning methods, which help augment the feature-extraction phenomenon for the process and enhance the quality of depth maps/accuracy of MDE (monocular depth estimation). Monocular depth estimation has seen improvements in many algorithms that can be deployed to give depth maps with better clarity and details around the edges and fine boundaries, which thus helps in delineating between thin structures. This paper reviews various recent deep learning-based stereo and monocular depth prediction techniques emphasizing the successes achieved so far, the challenges acquainted with them, and those that can be expected shortly.