Two dual frequency VLBI surveys of the compact structures in complete high-frequency samples of radio sources have been carried out. A classification of powerful extragalactic radio sources based on these surveys has revealed that similar to 10% of the objects are ''compact symmetric objects'' (CSOs) of an overall size less than or similar to 1 kpc, with high-luminosity lobes straddling the center of activity. In many respects these objects mimic Cygnus A, but are smaller by a factor of 10(2)-10(4). The working surfaces, or ''hot spots,'' of the jets in CSOs are much closer to the central engine than in other powerful radio sources, and they provide both a new laboratory for the study of the physics of relativistic jets and a unique probe of the interstellar medium in the range 1-500 pc from the central engines of active galaxies. The properties and statistics of CSOs are presented. The radio galaxy 2352+495 is shown to be a typical member of this class, and its optical and radio properties are discussed in detail. Our analysis of the hot spots in this object, in 0108+388, and in 0710$-439 shows that if these are confined by ram pressure then CSOs have ages much less than 10(6) yr. Two models for CSOs are investigated-a ''fast'' model in which the hot spots are advancing at similar to 0.02c and a ''slow'' model in which the velocity of advance is similar to 4 x 10(-3)c. On the ''slow'' model, the morphology of these objects would be more spherical than is observed, and there would likely be more reddening of the narrow emission lines relative to the galactic continuum emission, whereas all of the observations are consistent with the fast model, in which the speed of advance is comparable to the speed of advance of the lobes in Cygnus A and the ages of CSOs are in the range 10(3)-3 x 10(4) yr. This has interesting implications for their evolution which are discussed in a separate paper.