Observational evidence on the disappearance of comets is classified and tested for signatures of rapidly advancing decay. Of the several categories of phenomena, emphasis is on the dissipating comets, a group of objects that are observed to expand rapidly in dimensions, to fade, and eventually to disappear on the sky background on a time scale of several days to weeks. The tail can become the brightest part of the object and survive the head. The process appears to be triggered by an eruptive dust-emission event several weeks before the ultimate demise. Analogy is drawn between the physical changes experienced by the dissipating comets and by the secondary nuclei of the split comets. It is suggested that the nuclei of the dissipating comets are strongly nonspherical, structurally heterogeneous, and very poorly cemented. Detailed descriptions of the physical behavior of six dissipating comets (1897 III, 1913 VI, 1926 III, 1954 II, 1957 IX, and 1974 XV) are presented and both similarities and differences between these and the headless sungrazing comet 1887 I are summarized. Limited evidence on P/Westphal, the only known short-period dissipating comet, suggests the presence of persistent brightness fluctuations coupled with generally elevated activity one revolution prior to the dissipating event. Potential roles of nongravitational forces (causing the nucleus to precess) and brightness fluctuations in the process of comet dissipation are considered and a few comets exemplifying these phenomena are singled out. It is concluded that the dissipating comets could indeed be completely disintegrating into particulate debris while under observation and that the shattered mass can be made comparable with the expected masses of subkilometer-sized comets if the mass-distribution function of the fragments is relatively flat.