Termination of protein synthesis on the ribosome requires that mRNA stop codons are recognized with high fidelity. This is achieved by specific release factor proteins that are very different in bacteria and eukaryotes. Hence, while there are two release factors with overlapping specificity in bacteria, the single omnipotent eRF1 release factor in eukaryotes is able to read all three stop codons. This is particularly remarkable as it is able to select three out of four combinations of purine bases in the last two codon positions. With recently determined 3D structures of eukaryotic termination complexes, it has become possible to explore the origin of eRF1 specificity by computer simulations. Here, we report molecular dynamics free energy calculations on these termination complexes, where relative eRF1 binding free energies to different cognate and near-cognate codons are evaluated. The simulations show a high and uniform discrimination against the near-cognate codons, that differ from the cognate ones by a single nucleotide, and reveal the structural mechanisms behind the precise decoding by eRF1.