Many accelerator-based computers have demonstrated that they can be faster and more energy-efficient than traditional high-performance multi-core computers. Two types of programmable accelerators are available in high-performance computing: general-purpose accelerators such as GPUs, and customizable accelerators such as FPGAs, although general-purpose accelerators have received more attention. This chapter reviews the state-of-the-art and current trends of high-performance customizable computers (HPCC) and their use in Computational Science and Engineering (CSE). A top-down approach is used to be more accessible to the non-specialists. The “top view” is provided by a taxonomy of customizable computers. This abstract view is accompanied with a performance comparison of common CSE applications on HPCC systems and high-performance microprocessor-based computers. The “down view” examines software development, describing how CSE applications are programmed on HPCC computers. Additionally, a cost analysis and an example illustrate the origin of the benefits. Finally, the future of the high-performance customizable computing is analyzed.