The Sum Complex XA,k associated with a subset A of the cyclic group Zn and an integer 1≤k≤n, is the (k−1)-dimensional simplicial complex on the vertex set Zn whose maximal simplices are the sets σ⊂Zn of cardinality k such that ∑x∈σx∈A. Sum complexes may be viewed as high dimensional analogues of Cayley graphs over Zn and are relevant to a number of problems in topological combinatorics. In this talk, we shall describe the homology of sum complexes as well as some of their applications, including: Construction of high dimensional trees from sum complexes. Upper bounds on Betti numbers in terms of links, and nearly matching lower bounds via sum complexes. Uncertainty inequalities for the finite Fourier transform and their connections to the topology of sum complexes. The talk is based in parts on joint work with Nati Linial and Mishael Rosenthal and with Amir Abu-Fraiha.