Polarization is an important property of electromagnetic waves. It has shown the great application potential in optical fiber, radar, and satellite communications, where polarization has been exploited to be a promising means of enhancing channel capacity. The existing research has shown that, in theory, it has great potential to sixfold the channel capacity by using six co-located orthogonally polarized electric and magnetic dipoles. Wireless communications exploits the electromagnetic medium, which is also intrinsically polarization-sensitive. In the past decades, wireless communications have mainly exploited the dimensions of time, frequency, and space, and most wireless channels, technologies, and applications have been devised without explicit consideration of polarization. To keep up with the recent exponentially growing network capacity, as such, leveraging polarization for wireless communications is a logical next step. However, exploiting polarization in wireless communications involves many challenges that differentiate it from conventional polarization applications in terms of optical fiber, radar, and satellite communications due to complex depolarization effects, restriction on additional gain offered, and limited degrees of freedom in the polarization domain. These challenges have inspired recent advances, particularly related to polarization channels, polarization technologies (such as polarization-based modulation, polarization-based signal sensing, polarization-based orthogonal transmission, and polarization-based filtering), and their applications to emerging communication scenarios including energy-efficient communications, cognitive radio networks, and in-band full-duplex transmission. This paper thoroughly surveys these recent advances and presents the state-of-the-art research progress on each aspect. Furthermore, open research issues and challenges are discussed in order to provide perspectives for future research directions.