This chapter will focus on a specific variant of reservoir computing (RC) which is based on time-multiplexing the input and readout layers. It is often referred to as delay-based or time-delay RC, as it can easily be realized with a single physical systems subject to self-feedback and does not require a network of numerous coupled nonlinear units. In this time-multiplexed scheme, the nonlinear transformation and mixing of the input data is realized via the internal dynamics of the nonlinear physical system. Strongly nonlinear optical elements, for example semiconductor lasers subject to optical feedback, are promising candidates and offer fast data processing rates combined with the possibility of on-chip hardware realization. The time-delay of the feedback loop has a twofold purpose; it induces a complex response and provides short term memory. While the intrinsic dynamical nonlinearity can be exploited for the computing, it can also lead to intrinsic instabilities that prevent stable computing operation. We will discuss this in detail and elaborate how the bifurcation structure, i.e. the phase diagram in parameter space, can help to find good operation conditions for delay-based RC. Please also see the chapter by Stephan Wong, Doris E. Reiter and Sang Soon Oh for a machine learning based method to determine phase diagrams. We will also focus on the fact that the specific memory requirements for a given reservoir computing task, for example the prediction of a complex time series, are not unique and vary a lot between tasks, leading to different prerequisites for the dynamical system in use.