Consider two identical chaotic system that are started from virtually identical initial conditions. In a short time, they would be observed to diverge from one another. While it is true that, given ample passage of time, the systems will come within an arbitrarily close distance of one another, they would drift apart once again. In their fascinating paper, Pecora and Carroll have recently described a system of two Lorenz oscillators. They could be metaphorically described as a master and a slave system. The master system, undergoing chaos, drives a part of the slave system. Incredibly enough, the two systems remain in perfect synchronization. Further analysis of this system and its generalization will be presented. Based on the generalization, a method to design chaotically synchronous systems will be presented. Practical applications of this phenomenon will be discussed.