We describe the structure of a triconnected graph with the help of its decomposition by 3-cutsets. We divided all 3-cutsets of a triconnected graph into rather small groups with simple structure, called complexes. A detailed description of all complexes is presented. Moreover, we prove that the structure of a hypertree can be introduced on the set of all complexes. This structure gives us a complete description of the relative disposition of the complexes. Bibliography: 10 titles.