A quantum computer, if built, will be to an ordinary computer as a hydrogen bomb is to gunpowder, at least for some types of computations. Today no quantum computer exists, beyond laboratory prototypes capable of solving only tiny problems, and many practical problems remain to be solved. Yet the theory of quantum computing has advanced significantly in the past decade, and is becoming a significant discipline in itself. This article explains the concepts and basic mathematics behind quantum computers and some of the promising approaches for building them. We also discuss quantum communication, an essential component of future quantum information processing, and quantum cryptography, widely expected to be the first practical application for quantum information technology.