We shall describe the geometry of a certain class of PEL type Shimura varieties. In particular, we shall study the Newton polygon stratification of the reduction in positive characteristic p of these Shimura varieties. We shall show that each stratum can be described in terms of the products of some smooth varieties (we call Igusa varieties) with the reduced fibers of the pertinent Rapoport-Zink spaces, and of the action on them of a certain p-adic group (which depends on the stratum). As a result of this analysis, weobtain a description of the l-adic cohomology of the Shimura varieties in terms of the l-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces, in the appropriate Grothendieck group.