The Banach-Stone Theorem (respectly, Kadison Theorem) says that two abelian (respectively, general) C*-algebras are isomorphic as C*-algebras (respectively, JB*-algebras) if and only if they are isomorphic as Banach spaces. We are interested in using different structures to determine C*-algebras. Here, we would like to study the disjointness structures of C*-algebras and investigate if it suffices to determine C*-algebras. There are at least four versions of disjointness structures: zero product, range orthogonality, domain orthogonality and doubly orthogonality. In this thesis, we first study these disjointness structures in the case of standard operator algebras. Then we extend these results to general C*-algebras, namely, C*-algebras with continuous trace.