A number of the innovations used in studying singularities in commutative algebra have come from the study of tight closure and its test ideal in rings of equal characteristic. In replicating these results in rings of mixed characteristic, it has been useful to find closure operations that share key properties with tight closure. By studying the shared structure of common closure operations in commutative algebra, we show that many tight closure properties, in particular the structure of the test ideal, hold for a much larger set of closure operations, including (big Cohen-Macaulay) module closures and mixed characteristic closures. In this talk, I will describe the structures that these closure operations have in common and share some of the results on test ideals that have come out of this theory. Parts of this research are joint with subsets of Neil Epstein, Janet Vassilev, Felipe Pérez, and Zhan Jiang.