As technologies like the Internet, artificial intelligence, and big data evolve at a rapid pace, computer architecture is transitioning from compute-intensive to memory-intensive. However, traditional von Neumann architectures encounter bottlenecks in addressing modern computational challenges. The emulation of the behaviors of a synapse at the device level by ionic/electronic devices has shown promising potential in future neural-inspired and compact artificial intelligence systems. To address these issues, this review thoroughly investigates the recent progress in metal-oxide heterostructures for neuromorphic applications. These heterostructures not only offer low power consumption and high stability but also possess optimized electrical characteristics via interface engineering. The paper first outlines various synthesis methods for metal oxides and then summarizes the neuromorphic devices using these materials and their heterostructures. More importantly, we review the emerging multifunctional applications, including neuromorphic vision, touch, and pain systems. Finally, we summarize the future prospects of neuromorphic devices with metal-oxide heterostructures and list the current challenges while offering potential solutions. This review provides insights into the design and construction of metal-oxide devices and their applications for neuromorphic systems.