The volume, veracity, variability, and velocity of data produced from the ever-increasing network of sensors connected to Internet pose challenges for power management, scalability, and sustainability of cloud computing infrastructure. Increasing the data processing capability of edge computing devices at lower power requirements can reduce several overheads for cloud computing solutions. This paper provides the review of neuromorphic CMOS-memristive architectures that can be integrated into edge computing devices. We discuss why the neuromorphic architectures are useful for edge devices and show the advantages, drawbacks and open problems in the field of neuro-memristive circuits for edge computing.