Deep learning (DL) in magnetic resonance imaging (MRI) shows excellent performance in image reconstruction from undersampled k-space data. Artifact-free and high-quality MRI reconstruction is essential for ensuring accurate diagnosis, supporting clinical decision-making, enhancing patient safety, facilitating efficient workflows, and contributing to the validity of research studies and clinical trials. Recently, deep learning has demonstrated several advantages over conventional MRI reconstruction methods. Conventional methods rely on manual feature engineering to capture complex patterns and are usually computationally demanding due to their iterative nature. Conversely, DL methods use neural networks with hundreds of thousands of parameters and automatically learn relevant features and representations directly from the data. Nevertheless, there are some limitations to DL-based techniques concerning MRI reconstruction tasks, such as the need for large, labeled datasets, the possibility of overfitting, and the complexity of model training. Researchers are striving to develop DL models that are more efficient, adaptable, and capable of providing valuable information for medical practitioners. We provide a comprehensive overview of the current developments and clinical uses by focusing on state-of-the-art DL architectures and tools used in MRI reconstruction. This study has three objectives. Our main objective is to describe how various DL designs have changed over time and talk about cutting-edge tactics, including their advantages and disadvantages. Hence, data pre- and post-processing approaches are assessed using publicly available MRI datasets and source codes. Secondly, this work aims to provide an extensive overview of the ongoing research on transformers and deep convolutional neural networks for rapid MRI reconstruction. Thirdly, we discuss several network training strategies, like supervised, unsupervised, transfer learning, and federated learning for rapid and efficient MRI reconstruction. Consequently, this article provides significant resources for future improvement of MRI data pre-processing and fast image reconstruction.