In recent years, the scientific community has maintained substantial interest in neuromorphic computing systems based on memristive devices owing to their unique advantages. Although there is a wide variety of memristors, a device with optimal parameters for neuromorphic computing has not been developed yet. This article presents comprehensive overview of the principal types and parameters of memristive devices. Memristors operating through valence change, electrochemical metallization, phase change, ferroelectric tunneling, and spin-tunneling effects are considered. The key characteristics, benefits and limitations of each memristor type are analyzed. The main approaches to enhancing parameters of memristors are presented, including optimization of composition and structure of memristive devices, circuit architecture and algorithmic solutions.