Cardiovascular disease (CVD) is the number one cause of death in the United States and worldwide. Among the various CVDs, coronary artery disease (CAD) is the leading cause of death among both men and women. Of the various forms of CADs, atherosclerosis is the primary cause. To investigate these arterial diseases, numerical simulations of blood flow in the arteries using fluid-structure interactions (FSI) with the finite element method were performed. First, simulations were performed assuming the arterial walls are rigid, and then they were extended to deformable arteries where contraction and expansion of the arteries are considered. Moreover, this study also investigated the outcome of bypass surgeries involving end-to-side and end-to-end bypass anastomosis. To help understand the effect of various flow/material characteristics on these surgeries and related issues, numerical investigations on artery-graft bypass models were conducted. The primary objectives of this research were as follows: (1) to validate the numerical simulations with existing experimental data, (2) to differentiate the effect of Newtonian and non-Newtonian fluid flow considering three-dimensional rigid models of the artery, (3) to investigate the effect of arterial geometry using both steady and pulsatile flow cases, (4) to provide some indication of the occurrence of atherosclerosis while describing the hemodynamic parameters, (5) to determine the extent of interaction between blood flow and the elastic walls while performing numerical simulations on various arterial geometries with steady and pulsatile flow, (6) to investigate the outcome of bypass surgery (various cases) with natural and synthetic grafts, and (7) to determine the occurrence of intimal hyperplasia following bypass surgery. In the computations, the non-Newtonian behavior of blood was described using the Carreau-Yasuda model. Generally, good agreement between the numerical and experimental results was observed in the velocity profiles, whereas some discrepancies were found in wall shear stress (WSS) distributions. The regions of the artery models for both steady and pulsatile flow cases, with low wall shear stresses correspond to regions of the body that are more susceptible to atherosclerosis; or intimal hyperplasia for the case of bypass surgery were identified. It was also found that the geometry of the artery plays an important role in the development of atherosclerosis. The comparison between the simulations considering rigid arteries and deformable arteries showed a substantial increase in wall shear stresses for the rigid artery. In addition, it was observed that the calculated difference in shear stress between the simulations performed using rigid wall assumptions with that of deformable walls was in the range of 30 to 40 percent at the maximum shear stress location. Therefore, it was concluded that the deformation of the arterial wall cannot be neglected while performing blood flow simulations.