Magnetic resonance imaging (MRI) provides detailed structural information of the internal body organs and soft tissue regions of a patient in clinical diagnosis for disease detection, localization, and progress monitoring. MRI scanner hardware manufacturers incorporate various post-acquisition image-processing techniques into the scanner’s computer software tools for different post-processing tasks. These tools provide a final image of adequate quality and essential features for accurate clinical reporting and predictive interpretation for better treatment planning. Different post-acquisition image-processing tasks for MRI quality enhancement include noise removal, motion artifact reduction, magnetic bias field correction, and eddy electric current effect removal. Recently, deep learning (DL) methods have shown great success in many research fields, including image and video applications. DL-based data-driven feature-learning approaches have great potential for MR image denoising and image-quality-degrading artifact correction. Recent studies have demonstrated significant improvements in image-analysis tasks using DL-based convolutional neural network (CNN) techniques. The promising capabilities and performance of DL techniques in various problem-solving domains have motivated researchers to adapt DL methods to medical image analysis and quality enhancement tasks. This paper presents a comprehensive review of DL-based state-of-the-art MRI quality enhancement and artifact removal methods for regenerating high-quality images while preserving essential anatomical and physiological feature maps without destroying important image information. Existing research gaps and future directions have also been provided by highlighting potential research areas for future developments, along with their importance and advantages in medical imaging.