Three-body systems play unique roles in nuclear few-body physics because they are both physically non-trivial to test interaction theories quantitatively and computationally feasible to make exact calculations. In this talk, with a brief review of the Faddeev scheme for nuclear three-body systems in three-dimensional momentum space, some high performance computing (HPC) techniques for solving the Faddeev equations of the nuclear three-body system will be presented. The focus will be on the fast algorithm for evaluating three-body interactions and the regularization of the moving pole in the free three-body propagator, which are the critical elements to the performance and accuracy of the whole numerical approach. Some interesting physics results will be highlighted to demonstrate the importance of exact solutions. In regard to further boost the applicability and computing performance of the Faddeev scheme on emerging HPC architectures, some necessary algorithm and implementation improvements based on current state of the art will be discussed.