We present a scenario model of the population of close binaries in which the brighter component is a helium or a carbon-oxygen white dwarf. The other, dimmer component can be a helium white dwarf, a carbon-oxygen white dwarf, or a low-mass (<=0.3 Msolar) main-sequence star. The model takes into account the effects of observational selection related to the cooling of white dwarfs and the relative brightness of components. The total birthrate of such binaries in our Galaxy is estimated to be ~0.17 yr-1. In the model, in 63% of all cases, the dimmer component is also a white dwarf. In 82% of the systems that consist of two close white dwarfs, the brighter component is a helium white dwarf. This explains why, in at least seven out of the eight white dwarf pairs found in recent years, the primary is composed of helium.We estimate that close white dwarf pairs may constitute 1/9 of the total sample of observed white dwarfs. Systems that have a total mass exceeding 1.4 Msolar and in which the components are close enough to merge in a Hubble time may constitute ~1/40 of all close white dwarf pairs. This means that the sample of observed white dwarf pairs must be at least quadrupled before one may hope to find a hypothetical Type Ia supernova (SN Ia) precursor. The total number of SN Ia precursors is estimated to exceed the observable number by a factor of about 20. The merger frequency of close binary helium white dwarfs in the Galaxy is estimated to be ~0.02 yr-1. This number is consistent with the fact that, in the total sample of known white dwarf pairs, there are three in which the components are close enough to merge in a Hubble time. In one system (WD 1101+364), the merger may result in the formation of a helium subdwarf (nondegenerate helium star), and in two others (WD 2331+290 and WD 0957-666), probably in the formation of a hydrogen-deficient subgiant and possibly later of an R CrB star. The predicted merger frequency is also consistent with the fact that two out of seven white dwarfs selected for their low mass are apparently single, while five are in close binaries. Supported in part by the NSF grant AST 94-17156.