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In most wired and wireless systems, carrier tracking is an essential task that allows the receiver to precisely synchronize with the carrier of the incoming signal. Stringent carrier tracking requirements are imposed in systems that are sensitive to carrier mismatches, such as orthogonal frequency division multiplexing (OFDM), digital communication receivers employing high-order constellations, and terrestrial- or satellite-based positioning systems, just to mention a few. In the recent years, even more critical requirements are being imposed due to the emergence of new applications and services that are pushing traditional systems to operate in much more challenging conditions than the ones for which they were originally designed. The presence of severe fading, signal outages, abrupt phase changes and high user dynamics, are currently compromising the validity of well-known and long-established traditional carrier tracking techniques, thus calling for the development of new robust carrier tracking algorithms. In this paper, we provide a detailed survey on the five main strategies that can be adopted to cope with the technical challenges of robust carrier tracking. These strategies range from some basic optimizations of current tracking loops, to the use of Kalman filter-based architectures, or the application of innovative carrier tracking techniques based on particle filters or compressive sensing. We will also review some open-loop techniques, which are widely adopted in burst-mode communications receivers, as an alternative and potential candidate solution for robust carrier tracking in harsh conditions.