Sigma-Delta modulators (or noise shapers as they are also called) are extensively used for analogue-to-digital and digital-to-analogue data conversion (signal processing). Their dynamical behaviour can appear chaotic. I will explore this behaviour from the point of view of nonlinear dynamical systems analysis. To begin, the difference equation model of the sigma-delta modulator is introduced, and some basic results for bounded stability are obtained. The model is cast formally as a discrete dynamical system, and important continuity results allowing for a linear analysis are established. Drawing on this, I conduct a theoretical study of conditions for chaos or nonchaos using an adapted definition of Devaney's definition of chaos. This study is extended to the dithered system, in the context of allowing stochastic aspects in the model. I then introduce a stochastic formulation of the long-run dynamics, which is applied to give conditions for uniformly distributed error behaviour - conditions under which important consequences arise when dither is used to control the error statistics.