The theory of rigidity studies the uniqueness of realizations of graphs, i.e., frameworks. Originally motivated by structural engineering, rigidity theory nowadays finds applications in many important problems such as predicting protein flexibility, Computer-Aided Design, sensor network localization, etc. The present thesis treats a wide range of problems concerning different kinds of rigidity, corresponding to different scopes of uniqueness (local/infinitesimal, global and universal), in various types of frameworks. First, we develop results in inductive construction and decomposition of graphs with mixed sparsity conditions as well as results on the packing of arborescences with matroidal constraints. These results are then used to obtain characterizations of infinitesimal rigidity in frameworks with mixed constraints. We also investigate the effect of extension operations on frameworks and extend a known result on the global rigidity preservation of 1-extension on direction-length frameworks in dimension two to all dimensions. For universal rigidity, where little is known, we obtain a complete characterization for the class of complete bipartite frameworks on the line. We also generalize a sufficient condition for the universal rigidity of frameworks by allowing non-general positions.
