The potential energy surfaces (PES) of the primary addition reactions of atomic oxygen O(D-1) and O(P-3) on acetylene were done using Density Functional Theory with the standard B3LYP hybrid functional. The crucial areas of these PES were then refined using the UMP2 and CCSD(T) schemes and, in few cases, CASPT2, MCSCF, and MR-AQCC ones. The stability of these primary products was studied. The reactivity of C2H2 and O, especially in low-temperature matrix conditions, is discussed on these bases. The singlet surface shows that the insertion of the oxygen into the C-H bond, resulting in ethynol HCdropCOH, is the main reaction. An addition onto the CdropC bond, leading to oxirene and/or formylcarbene HC-CHO is nevertheless to be considered. A particular attention was given to the oxirene-formylcarbene region and MR-AQCC calculations predict an extremely flat area in agreement with previous CCSD(T) results. Secondary transposition of formylcarbene into ketene H2C=C=O could be avoided at very low temperature. From the triplet state of the oxygen atom, the triplet formylcarbene is expected as the major product. Nevertheless, a cyclic nonplanar species, the triplet state of oxirene, is stable on the triplet surface and could be trapped at low temperature, provided an intersystem triplet singlet crossing does not occur, which would lead to singlet oxirene, if stable. If not, this process would lead to singlet formyl carbene and finally to its triplet ground state. Indeed, singlet and triplet surfaces are very close to one another in this part of the PES. Hydrogen transposition giving triplet ketene can be ruled out in matrix conditions. The formation of cyclic three-member carbenes -C-O-CH2-, as well singlet as triplet, is unlikely though both species are rather stable on their respective PES.