The application of biologically inspired methods in design and control has a long tradition in robotics. Unlike previous approaches in this direction, the emerging field of neurorobotics not only mimics biological mechanisms at a relatively high level of abstraction but employs highly realistic simulations of actual biological nervous systems. Even today, carrying out these simulations efficiently at appropriate time-scales is challenging. Neuromorphic chip designs specially tailored to this task therefore offer an interesting perspective for neurorobotics. Unlike standard Von Neumann CPUs, these chips can not be simply programmed with a standard programming language. Like real brains, their functionality is determined by the structure of neural connectivity and synaptic efficacies. Enabling higher cognitive functions for neurorobotics consequently requires the application of neurobiological learning algorithms to adjust synaptic weights in a biologically plausible way. In this paper, we therefore investigate how to program neuromorphic chips by means of learning. First, we provide an overview over available neuromorphic chip designs and analyze them in terms of neural computation, communication systems and software infrastructure. On the theoretical side, we review available neurobiological learning techniques. Based on this overview, we then examine on-die implementations of these learning algorithms on the considered neuromorphic chips. A final discussion puts the findings of this work into context and highlights how neuromorphic hardware can potentially advance the field of autonomous robot systems. The paper thus gives an in-depth overview of neuromorphic implementations of basic mechanisms of synaptic plasticity which are required to realize advanced cognitive capabilities with spiking neural networks.