The authors introduce a new combinatorial structure whose definition is based on a generalization of the Deza-Erdős-Frankl result on set systems with prescribed intersection sizes. They first prove the generalized theorem, and then define squashed geometries, and squashed designs as extremal configurations for this result. These new structures include t-designs, matroids, geometries transversal designs and other well known configurations as special cases. The paper contains many interesting examples, and a wealth of open problems. The style is terse, but the content justifies the time invested in a careful study of this excellent paper.