This paper reports CCD observations of cometary comae in the spectral continuum. Radial brightness profiles of 14 comets and isophotal maps of three are presented. For 10 of the 14, the azimuthally averaged brightness in the vicinity of the nucleus falls off faster than rho--1, where rho is the projected radial distance from the nucleus in the plane of the sky. With the aid of simple modeling, we show that radiation pressure alone cannot account for the observed departures from rho--1, so we postulate that the grains fade (i.e., decrease in albedo or size) while they flow outward. For only three of the comets [Austin (1984 XIII), Hartley-Good (1985 XVII), and P/Johnson] was there no evidence of any grain fading at all. Fading was also undetected in Comet Bowell (1982 I) at 8.58 AU, but was found in Bowell at 5.98 AU. Cases with the most rapid grain fading (P/Neujmin 1, P/Arend-Rigaux, P/Churyumov-Gerasimenko, P/IRAS, and P/Giacobini-Zinner) occurred at small heliocentric distances, but so did some of the cases with no fading, thus indicating large intrinsic differences from comet to comet in the nature of the grain population. The other observed comets were Cernis (1983 XII), P/Hartley-IRAS, P/Russell 4, P/Smirnova-Chernykh, and P/Takamizawa.