Memristors are considered to be the potential candidate for simulating synapses due to their high density, low power consumption and continuously adjustable resistance. Metal oxide is an ideal choice for fabricating memristive devices with high performance due to its advantages of oxygen migration, easy adjustment of components and compatibility with traditional CMOS. In this review paper, the memristive behaviors and operation mechanism of oxide-based memristors including digital-type memristors and analog-type memristors are first introduced. We mainly summarize the cognitive functions simulated by analog-type memristive synapse, including nonlinear-transmission characteristic, synaptic plasticity, learning experience, and non-associative/associative learning. Then, the potential applications of memristive synapse in pattern recognition, sound localization, logic operation, flexibility/transferability and optoelectronic memristive synapse are introduced. Finally, we provide an outlook of the future possible studies of oxide-based memristive synapse in the relevant fields.