  The Poincar\'e-Bendixson theorem plays an important role in the study of the
qualitative behavior of dynamical systems on the plane; it describes the
structure of limit sets in such systems. We prove a version of the
Poincar\'e-Bendixson Theorem for two dimensional hybrid dynamical systems and
describe a method for computing the derivative of the Poincar\'e return map, a
useful object for the stability analysis of hybrid systems. We also prove a
Poincar\'e-Bendixson Theorem for a class of one dimensional hybrid dynamical
systems.
