The discovery of the channel polarization phenomenon and polar codes is one of the most recent fundamental advances in coding theory. Polar codes provably achieve the fundamental limit of capacity for a wide range of channels with explicit constructions and low encoding and decoding complexity. In the first part of this talk, we provide a brief overview of polar codes and describe the most recent advances towards incorporating them in 5G wireless communications. In the second part, multiple-access communication channels are considered and we show how to efficiently achieve the fundamental capacity limits in this case using polar codes. In the last part, we discuss some of the recent advances in characterizing the finite-length behavior of polar codes.
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