The nearly epidemic expansion of cardiovascular diseases in western societies consist a serious threat to public health. In the last few decades, the proliferation of cardiovascular events along with the perpetual increase of life expectancy, generate the need for novel methodologies and tools in order to assist diagnostic and therapeutic management and design. The investigation of biochemical and mechanical factors within complex biological systems, such as encountered in coronary networks, merges multidisciplinary scientific fields. Although the mechanisms for the initiation and progression of atherosclerosis are not fully understood, it is well accepted that local blood flow dynamics play a major role in the localization of coronary artery disease (CAD). It has been shown that the most important risk factors for the development of CAD are low and/or high wall shear stress (WSS) regions, high pressure distributions along the vessels and long residence times of blood cells and lipoids at specific sites in vasculature. Moreover, in vivo animal models have revealed that specific sites in vascular network, such as heel, toe, suture line and walls just opposite to the junction, are most prone to CAD. The concept of total arterial revascularization (TAR) is widely accepted in modern practice of cardiac surgery, aiming at bypassing pathological conduits. This has led to the construction of composite arterial coronary grafts (CACGs) consisting mainly from different segments of internal thoracic and radial arteries. CACGs, in the routine practice of cardiac surgery, are conjugated to function either in parallel as Y, T and Π or in series as sequential graft, according to human anatomy. However, it is still unknown whether or not these CACGs can adequately meet the metabolic demands of the myocardium, especially under different loading conditions of the heart (coronary flow reserve). In the present work, a mathematical model for the simulation of blood flow within different configurations of CACGs under both steady and unsteady flow conditions is developed. The computational grids were generated based on multi-block structure in commercial CFD code, namely PHOENICS. The three-dimensional equations of fluid motion are numerically solved in a curvilinear, non-orthogonal coordinate system. The flow field is discretized by the Finite Volume Method and the solution is obtained using a guess-and-correct procedure employing a modified version of the semiimplicit method for pressure-linked equations (SIMPLEST) algorithm. The numerical results of simulated blood flow within CACGs focus on computation of critical haemodynamic parameters such as flow rate ratios, velocity and WSS distributions and pressure drop along the arterial vessels. Furthermore, the spatial and temporal variations of haemodynamic factors under pulsatile flow conditions are numerically computed. The computational results are found in excellent agreement with in vitro experimental measurements of fluid (mixture of glycerol and water) flow for steady state conditions.