Aims: We present the first simulations of the full evolution of super-AGB stars through the entire thermally pulsing AGB phase. We analyse their structural and evolutionary properties and determine the first SAGB yields. Methods: Stellar models of various initial masses and metallicities were computed using standard physical assumptions which prevents the third dredge-up from occurring. A postprocessing nucleosynthesis code was used to compute the SAGB yields, to quantify the effect of the third dredge-up (3DUP), and to assess the uncertainties associated with the treatment of convection. Results: Owing to their massive oxygen-neon core, SAGB stars suffer weak thermal pulses, have very short interpulse periods and develop very high temperatures at the base of their convective envelope (up to 140 ?? 108 K), leading to very efficient hot bottom burning. SAGB stars are consequently heavy manufacturers of 4He, 13C, and 14N. They are also able to inject significant amounts of 7Li, 17O, 25Mg, and 26,27Al in the interstellar medium. The 3DUP mainly affects the CNO yields, especially in the lower metallicity models. Our post-processing simulations also indicate that changes in the temperature at the base of the convective envelope, which would result from a change in the efficiency of convective energy transport, have a dramatic impact on the yields and represent another major source of uncertainty.