Hybrid systems are dynamical systems where the state is allowed to both flow continuously and jump discretely. This dissertation addresses three somewhat disparate problems that are united by using hybrid system tools for their resolution. The first topic

is a parameter estimation algorithm where an existing algorithm is embedded within a hybrid system, thereby permitting the use of several relevant tools. It is shown that a set corresponding to correct parameter estimation has a robust stability property and

that a persistency of excitation condition leads to convergence to the correct parameter estimate in finite time. The second topic involves implementing hybrid control through high-gain observers. Here results on robustness of stability are repeatedly exploited on

the way to providing a semi-global practical stability result. Finally, the third topic is a distributed algorithm for synchronizing agents on a circle. Here a hybrid system approach is necessary to overcome the topological obstruction of being confined to a circle,

while randomness is needed to overcome symmetry issues.