Image reconstruction is the process of generating an image of an object from the signals captured by the scanning machine. Medical imaging is an interdisciplinary field combining physics, biology, mathematics and computational sciences. This paper provides a complete overview of image reconstruction process in MRI (Magnetic Resonance Imaging). It reviews the computational aspect of medical image reconstruction. MRI is one of the commonly used medical imaging techniques. The data collected by MRI scanner for image reconstruction is called the k-space data. For reconstructing an image from k-space data, there are various algorithms such as Homodyne algorithm, Zero Filling method, Dictionary Learning, and Projections onto Convex Set method. All the characteristics of k-space data and MRI data collection technique are reviewed in detail. The algorithms used for image reconstruction discussed in detail along with their pros and cons. Various modern magnetic resonance imaging techniques like functional MRI, diffusion MRI have also been introduced. The concepts of classical techniques like Expectation Maximization, Sensitive Encoding, Level Set Method, and the recent techniques such as Alternating Minimization, Signal Modeling, and Sphere Shaped Support Vector Machine are also reviewed. It is observed that most of these techniques enhance the gradient encoding and reduce the scanning time. Classical algorithms provide undesirable blurring effect when the degree of phase variation is high in partial k-space. Modern reconstructions algorithms such as Dictionary learning works well even with high phase variation as these are iterative procedures.