There is good reason to suspect that changes in the housing market had something to do with the dramatic rise in the homelessness in the United States after 1980. Whether the housing market is responsible, however, remains a controversial question. Resolving this question takes two things: a theory of the housing market that includes homelessness and relates it with other measurable phenomena, and a great deal of serious empirical work to test the implications of that theory. This paper concentrates on the first task. The goal is to provide a fully worked-out model of the housing market with the possibility of homelessness. The basic set-up is a Sweeney model of filtering with a continuum of qualities of housing.