Coboundary expansion and cosystolic expansion are generalizations of edge expansion to hypergraphs. In this talk, we will first explain how the generalizations work. Next we will motivate the study of such hypergraphs by looking at their applications to property testing. Finally, we will illustrate the use of the cone method to show coboundary expansion for the complete complexes. Time permitting, we will also sketch the local-to-global argument for showing cosystolic expansion.