In this thesis we study piecewise smooth and switched positive systems and investigate the monotonicity properties of such systems. We describe many examples of such systems, particulary drawing from the mathematical biology literature, in order to motivate the work in later chapters. We describe the mathematical theory behind our work in Chapters 3 and 4. In particular we review the theory of LTI systems, positive LTI systems and monotone systems, indicating how monotonicity can be used to determine the asymptotic behaviour of positive LTI systems. We also discuss issues which arise in the study of piecewise smooth and switched linear systems and review solution concepts for such systems. In Chapter 5, we extend the Kamke conditions for smooth monotonic systems to piecewise smooth systems, and in certain cases show that they are equivalent to the monotonicity of the system.