At the electrode–electrolyte interface, hydrogen shows peculiar behavior being influenced by the dipole layer called Helmholtz layer. For the understanding of this system, first-principles methods have been intensively developed and applied to the benchmarking interface like Pt(111)-water; however, controversy still exists even on the hydrogen adsorption although being the simplest and most-studied problem. After giving introductory explanation of the electrochemical interface and reaction, we briefly review the advance of the theory. It has been revealed that the typical density functional calculations are occasionally not accurate enough and need to advance the description of electron correlation and nuclear quantum effect. The advanced calculation shows a promise of elucidating some... experimental data providing therewith further theoretical challenges.