The genetic code is a set of instructions that determine how the information in our genetic material is translated into amino acids. In general, it is universal for all organisms, from viruses and bacteria to humans. However, in the last few decades, exceptions to this rule have been identified both in pro- and eukaryotes. In this review, we discuss the 16 described alternative eukaryotic nuclear genetic codes and observe theories of their appearance in evolution. We consider possible molecular mechanisms that allow codon reassignment. Most reassignments in nuclear genetic codes are observed for stop codons. Moreover, in several organisms, stop codons can simultaneously encode amino acids and serve as termination signals. In this case, the meaning of the codon is determined by the additional factors besides the triplets. A comprehensive review of various non-standard coding events in the nuclear genomes provides a new insight into the translation mechanism in eukaryotes.