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The emerging fifth generation (5G) wireless communication system raises new requirements on spectral efficiency and energy efficiency. A massive multiple-input multiple-output (MIMO) system, equipped with tens or even hundreds of antennas, is capable of providing significant improvements to spectral efficiency, energy efficiency, and robustness of the system. For the design, performance evaluation, and optimization of massive MIMO wireless communication systems, realistic channel models are indispensable. This article provides an overview of the latest developments in massive MIMO channel measurements and models. Also, we compare channel characteristics of four latest massive MIMO channel models, such as receiver spatial correlation functions and channel capacities. In addition, future challenges and research directions for massive MIMO channel measurements and modeling are identified.