Memristors have been intensively studied in recent years as potential building blocks for the construction of versatile neuromorphic architectures. The prevalent developments focus on nanoionic devices due to their ease of reversible regulation, non-volatility, scalability down to the atomic level, low-energy consumption, nanosecond response times, sufficient yield and reliability, as well as their homologous regulatory mechanisms with biological synapses. However, despite their desirable structures and excellent properties, the previously reported properties of nanoionic devices are rather diverse and dissatisfactory for the delivery of reliable analog properties, which is the precondition for the imitation of brain-like functions. Here, the general requirements of neuromorphic engineering in terms of device structure, characterization parameters, synaptic functions and device engineering are introduced. A critical overview of the proposed nanoionic mechanisms for memristive switching is given, focusing particularly on providing fundamental insights into the strategies for regulating the adaptive memristive characteristics of devices that resemble the behaviors of biological synapses, which is an element of neural networks. In addition, the research progress in active materials (e.g., oxides, polymers, small molecules, bio-macromolecules and dopants), especially regarding the correlation among materials, is elaborated and a rational approach to provide insight into new design strategies is considered, and how to understand the memristor mechanisms and further achieve excellent memristive devices is discussed. Finally, the current challenges and several possible future research directions in this area are discussed.