In vivo magnetic resonance imaging (MRI) has been widely used in the clinical setting and radiological research. Since it is radiation free and can provide images of high quality, it has played a promising and increasingly active role in medicine overall. Most clinical MRI applications collect signal from protons, either as part of water or fat. By applying different imaging schemes and acquisition times, MRI can provide excellent contrast between soft tissues based on whether the signal is from water protons or fat ones. Due to the difference between biochemical structures of fat and water, fat signal can appear bright in many important imaging scenarios and therefore obscure underlying pathology such as inflammation, edema or progression of certain tumor. Meanwhile, direct visualization of fat distribution is also desirable in various pathologies such as fatty tumors, including adrenal adenomas, angiomyolipomas and other fat-related mesenchymal tumors. Quantification of visceral adipose tissue is also enjoying a great interest in recent studies. In these cases, separation of water and fat signals can be highly useful by providing water-only and/or fat-only images. However, the separation of fat and water signals in MRI remains difficult due to multiple hurdles. The mathematical model for the fat water signal is flawed by nature in the presence of fat-or water-only locations. On the other hand, while technology advancement in the MRI scanner is bringing more promising diagnosis and treatment plans for patients, it also poses challenges in redesigning current fat water separation methods for the advancement. Large B0 magnetic field variations vii can make the separation problem more ill-conditioned. The increasing data size and resolution demand faster, simpler and more accurate approaches to solve the fat water separation problem. The goal of this thesis is to develop new fat water separation methods by employing graph search based models. The developed methods achieve global optimality of the newly designed problem formulations for fat water separation. New frameworks are also proposed in this work to reduce the computational time by an order of magnitude compared to state-of-the-art methods. With experiments on a variety of datasets, particularly with challenging anatomical regions and of large size, the proposed methods have demonstrated their great potential to improve the stateof-the-art methods in more applications of MRI fat water separation. viii