The symmetry of the wavefunction describing the Cooper pairs is one of the most fundamental quantities in a superconductor, but for iron-based superconductors it has proved to be problematic to determine, owing to their complex multi-band nature [1][2][3] . Here we use a first-principles many-body method, including the two-particle vertex function, to study the spin dynamics and the superconducting pairing symmetry of a large number of iron-based compounds. Our results show that these high-temperature superconductors have both dispersive high-energy and strong low-energy commensurate or nearly commensurate spin excitations, which play a dominant role in Cooper pairing. We find three closely competing types of pairing symmetries, which take a very simple form in the space of active iron 3d orbitals, and di er only in the relative quantum mechanical phase of the xz, yz and xy orbital components of the Cooper pair wavefunction. The extensively discussed s The spin and the multi-orbital dynamics of iron-based superconductors are believed to play an essential role in the mechanism of superconductivity 6 , but a realistic modelling of magnetic excitations, and a clear physical picture for their variation across different families of iron superconductors, is currently lacking. The Cooper pairs are locked into singlets, but the orbital structure of the superconducting order parameter can be material dependent, and its connection to orbital and spin excitations is an open problem.The charge dynamics of the iron-based superconductors is controlled by the strong Hund's coupling on the iron site 7,8 , which requires a theoretical approach that simultaneously treats the itinerancy of the electrons and Hund's interaction on an equal footing. Using non-perturbative many-body method and ab initio-determined two-particle scattering amplitude (the twoparticle vertex function), we are able to accurately describe the spin dynamics and symmetry of the superconducting order parameter, and we will show that Hund's rule coupling and orbital blocking 9 play a crucial role in the superconductivity of iron superconductors.All iron-based superconductors contain the same basic motiflayers of iron atoms tetrahedrally coordinated by pnictogen or chalcogen atoms-but their spin excitation spectra varies greatly among compounds. In Fig. 1 we plot the dynamic spin structure factor S(q, ω) = χ (q,ω)/(1 − exp(−hω/k B T )) in the paramagnetic state for several classes of iron compounds along the high-symmetry momentum path in the first Brillouin zone of the single-iron unit cell. Here, the momentum transfer is labelled using the same convention as used in neutron scattering experiments 10 . We overlay the neutron scattering data 10-13 for some compounds where experimental results are available, to show the good agreement between theory and experiment. We computed the magnetic excitations in the paramagnetic state, at temperatures above the spin density wave (SDW) transition (even for compounds that have a magnetically ordered ground state) and compa...