With the end of Moore's law in sight, new computing paradigms are needed to fulfill the increasing demands on data and processing potentials. Inspired by the operation of the human brain, from the dimensionality, energy and underlying functionalities, neuromorphic computing systems that are building upon circuit elements to mimic the neurobiological activities are good concepts to meet the challenge. As an important factor in a neuromorphic computer, electronic synapse has been intensively studied. The utilization of transistors, atomic switches and memristors has been proposed to perform synaptic functions. Memristors, with several unique properties, are exceptional candidates for emulating artificial synapses and thus for building artificial neural networks. In this paper, metal oxide-based memristor synapses are reviewed, from materials, properties, mechanisms, to architecture. The synaptic plasticity and learning rules are described. The electrical switching characteristics of a variety of metal oxide-based memristors are discussed, with a focus on their application as biological synapses.