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Space exploration has been on the rise since the 1960s. Along with the other planets such as Mercury, Venus, Saturn, and Jupiter, Mars certainly plays a significant role in the history of space exploration and has the potential to be the first extraterrestrial planet to host human life. In this context, tremendous effort has been put into developing new technologies to photograph, measure, and analyze the red planet. As the amount of data collected from science instruments around and on Mars increased, the need for fast and reliable communication between Earth and space probes has emerged. However, communicating over deep space has always been a big challenge due to the propagation characteristics of radio waves. Nowadays, the collaboration of private companies like SpaceX with space agencies to make Mars colonization a reality, introduces even more challenges, such as providing high data rate, low latency, energy-efficient, reliable, and mobility-resistant communication infrastructures in the Martian environment. Propagation medium and wireless channel characteristics of Mars should be extensively studied to achieve these goals. This survey article presents a comprehensive overview of the Mars missions and channel modeling studies of the near-Earth, interstellar, and near-planet links. Studies featuring three-dimensional (3D) channel modeling simulations on the Martian surface are also reviewed. We have also presented our own computer simulations considering various scenarios based on realistic 3D Martian terrains using the Wireless Insite software. Path loss exponent, power delay profile, and root-mean-square delay spread for these scenarios are calculated and tabularized in this study. Furthermore, future insights on emerging communication technologies for Mars are given.