January, 1993. This paper consists of three more or less independent parts. The first one contains a construction which produces a tensor triangulated category (called the homological category) out of a site with some additional structure. In the second one we define two new Grothendieck topologies on the category of Noetherian schemes called h- and qfh-topology and prove some of their basic properties including the comparison theorems for cohomology. Finally in the third part we apply the construction of homological category to the category of schemes over a base S considered as a site with respect to h- or qfh-topology. We show that the resulting tensor triangulated category satisfies many of the properties on would expect from the derived category of mixed motives over S. Since this paper was written, a much better understanding of the structure of this triangulated category in the case when the base scheme is the spectrum of a field has been achieved. See "Triangulated categories of motives over a field".