Abstract In the past 10 years, neuromorphic computing has emerged as a novel approach to tackle the challenges posed by the end of Moore’s law. Memristive devices are very promising due to their unique properties. They are highly compact, fast switching, power efficient, and can represent multiple states of memory (via a tunable resistance). As such they can find use as synaptic connections among neurons in biologically inspired hardware neural networks, and therefore could be a critical element for the development of hardware cognitive systems with capabilities such as those found in animal nervous systems. In this chapter, we present an overview on the current status of synaptic circuits based on memristive devices. We review various implementations including the single-memristor synapse, which employs resistive switching random access memory, phase-change memory and spin-transfer torque magnetic random access memory, hybrid structures combining complementary metal-oxide semiconductor transistors, and memristive devices and materials-based approaches aiming at reproducing biological learning rules by the physical properties of the device. Learning rules such as the spike-timing-dependent plasticity, the spike-rate-dependent plasticity and the short-term plasticity are described. We finally present some examples of learning circuits exploited in hardware neural networks, which make initial steps on a path toward memristive circuits capable of biorealistic brain-inspired cognitive computing.