Several analog and digital brain-inspired electronic systems have been recently proposed as dedicated solutions for fast simulations of spiking neural networks. While these architectures are useful for exploring the computational properties of large-scale models of the nervous system, the challenge of building low-power compact physical artifacts that can behave intelligently in the real-world and exhibit cognitive abilities still remains open. In this paper we propose a set of neuromorphic engineering solutions to address this challenge. In particular, we review neuromorphic circuits for emulating neural and synaptic dynamics in real-time and discuss the role of biophysically realistic temporal dynamics in hardware neural processing architectures; we review the challenges of realizing spike-based plasticity mechanisms in real physical systems and present examples of analog electronic circuits that implement them; we describe the computational properties of recurrent neural networks and show how neuromorphic Winner-Take-All circuits can implement working-memory and decision-making mechanisms. We validate the neuromorphic approach proposed with experimental results obtained from our own circuits and systems, and argue how the circuits and networks presented in this work represent a useful set of components for efficiently and elegantly implementing neuromorphic cognition.