We present new theoretical stellar yields and surface abundances for asymptotic giant branch (AGB) models with a metallicity appropriate for stars in the Small Magellanic Cloud (SMC, Z = 0.0028, [Fe/H] ≈ −0.7). New evolutionary sequences and post-processing nucleosynthesis results are presented for initial masses between 1 and 7  M_⊙, where the 7  M_⊙ is a super-AGB star with an O–Ne core. Models above 1.15  M_⊙ become carbon rich during the AGB, and hot bottom burning begins in models M ≥ 3.75 M_⊙. We present stellar surface abundances as a function of thermal pulse number for elements between C to Bi and for a selection of isotopic ratios for elements up to Fe and Ni (e.g. ¹²C/¹³C), which can be compared to observations. The integrated stellar yields are presented for each model in the grid for hydrogen, helium, and all stable elements from C to Bi. We present evolutionary sequences of intermediate-mass models between 4 and 7  M_⊙ and nucleosynthesis results for three masses (M = 3.75, 5, and 7 M_⊙) including s-process elements for two widely used AGB mass-loss prescriptions. We discuss our new models in the context of evolved AGB and post-AGB stars in the SMCs, barium stars in our Galaxy, the composition of Galactic globular clusters including Mg isotopes with a similar metallicity to our models, and to pre-solar grains which may have an origin in metal-poor AGB stars.