Functional and scalable memory selector devices are essential for high-density memory and storage. This paper reviews the performance requirements and device options of two-terminal memory selectors for crossbar arrays. In a large crossbar array without appropriate selector devices, large number of sneak paths will significantly degrade the reading signal and writing conditions. Asymmetry and nonlinearity in selector device characteristics can both improve crossbar array operation by making sneak paths more resistive. Rectifying diodes, nonlinear devices, and volatile switches can provide basic selector functions; however, they also need to be balanced with memory elements and meet rigorous requirements in large arrays. Modeling plays an important role in the assessment of selector device function and crossbar array performance. This paper will review the selector device and crossbar array modeling approaches and summarize key observations. The design of large crossbar arrays with functional selector devices requires a comprehensive approach that incorporates device characteristics, array parameters, operation conditions, and application specifications.