Unmanned aerial vehicles (UAVs) have attracted great interest in rapid deployment for both civil and military applications. UAV communication has its own distinctive channel characteristics compared to the widely used cellular or satellite systems. Accurate channel characterization is crucial for the performance optimization and design of efficient UAV communication. However, several challenges exist in UAV channel modeling. For example, the propagation characteristics of UAV channels are under explored for spatial and temporal variations in non-stationary channels. Additionally, airframe shadowing has not yet been investigated for small size rotary UAVs. This paper provides an extensive survey of the measurement methods proposed for UAV channel modeling that use low altitude platforms and discusses various channel characterization efforts. We also review from a contemporary perspective of UAV channel modeling approaches, and outline future research challenges in this domain.