Many decades after quantum mechanics was accepted as a more accurate model of the physical world, Richard Feynman proposed the brilliant vision of building computers based on quantum mechanical systems in 1981. The two major drivers for the development in this field are the need to overcome the limitations both in the theory of computing with classical deterministic digital computing models and in the technology of building them. With a brief introduction to the motivation for quantum computing, we will discuss the basic model and the operations. Notwithstanding the limitations in physical realizations, the progress in the technology of quantum computers during the last quarter of a century is sketched very briefly. The main focus is on synthesis methodology for quantum computers at the logical level starting with a set of fundamental quantum gates.