James, Matthew R.These lecture notce introduce some basic concepts concerning smooth manifolds and Lie groups, since they are often the natural mathematical setting for studying nonlinear estimation and control problems. After reviewing some multivariable calculus, we discuss manifolds, derivative maps, vector fields and flows. Then Lie groups and Lie algebras are defined and discussed. Finally, we discuss the paper by Lo and Willsky which treats nonlinear estimation problems on the circle. A number of examples and illustrations are provided.en-USTechnical Report