Magnetic resonance imaging (MRI) is one of the most prevalent imaging modalities used for diagnosis, treatment planning, and outcome control in various medical conditions. MRI sequences provide physicians with the ability to view and monitor tissues at multiple contrasts within a single scan and serve as input for automated systems to perform downstream tasks. However, in clinical practice, there is usually no concise set of identically acquired sequences for a whole group of patients. As a consequence, medical professionals and automated systems both face difficulties due to the lack of complementary information from such missing sequences. This problem is well known in computer vision, particularly in medical image processing tasks such as tumor segmentation, tissue classification, and image generation. With the aim of helping researchers, this literature review examines a significant number of recent approaches that attempt to mitigate these problems. Basic techniques such as early synthesis methods, as well as later approaches that deploy deep learning, such as common latent space models, knowledge distillation networks, mutual information maximization, and generative adversarial networks (GANs) are examined in detail. We investigate the novelty, strengths, and weaknesses of the aforementioned strategies. Moreover, using a case study on the segmentation task, our survey offers quantitative benchmarks to further analyze the effectiveness of these methods for addressing the missing modalities challenge. Furthermore, a discussion offers possible future research directions.