This paper reviews the nucleosynthesis in asymptotic giant branch (AGB) stars, focusing on the development of theoretical models and their relationship to observations. The review highlights the importance of high-resolution codes with improved opacities, which have successfully accounted for the third dredge-up. This allows for a better understanding of low-luminosity C stars, which are enriched in s-elements, as a normal outcome of AGB evolution. The production of 12C and neutron-rich nuclei in the He intershell and mass loss from strong stellar winds are key processes. Neutron captures in AGB stars are driven by two reactions: 13C(α,n)16O and 22Ne(α,n)25Mg, which operate at different conditions. The first reaction occurs in the radiative interpulse phase, while the second is activated during convective thermal pulses. These processes result in complex nucleosynthesis phenomena that cannot be approximated analytically. Theoretical models at different metallicities account for observational constraints from evolved red giants, unevolved stars, presolar grains, and solar system s-process isotopes. The solar s-process pattern is not a standard archetypal s-process but is the result of Galactic chemical evolution mixing the outputs of AGB stars with different metallicities. The paper also discusses the possibility that a close encounter with an AGB star may have contributed to the short-lived radioactivity in the early solar system, such as 26Al, 41Ca, 60Fe, and 107Pd. The existence of these radioactivities suggests that a nearby AGB star polluted the protosolar nebula with a small amount of neutron exposure and a low initial mass. The paper concludes by emphasizing the need for further hydrodynamical studies to address the limitations in current mixing algorithms and the role of AGB stars in galactic enrichment and solar system formation.This paper reviews the nucleosynthesis in asymptotic giant branch (AGB) stars, focusing on the development of theoretical models and their relationship to observations. The review highlights the importance of high-resolution codes with improved opacities, which have successfully accounted for the third dredge-up. This allows for a better understanding of low-luminosity C stars, which are enriched in s-elements, as a normal outcome of AGB evolution. The production of 12C and neutron-rich nuclei in the He intershell and mass loss from strong stellar winds are key processes. Neutron captures in AGB stars are driven by two reactions: 13C(α,n)16O and 22Ne(α,n)25Mg, which operate at different conditions. The first reaction occurs in the radiative interpulse phase, while the second is activated during convective thermal pulses. These processes result in complex nucleosynthesis phenomena that cannot be approximated analytically. Theoretical models at different metallicities account for observational constraints from evolved red giants, unevolved stars, presolar grains, and solar system s-process isotopes. The solar s-process pattern is not a standard archetypal s-process but is the result of Galactic chemical evolution mixing the outputs of AGB stars with different metallicities. The paper also discusses the possibility that a close encounter with an AGB star may have contributed to the short-lived radioactivity in the early solar system, such as 26Al, 41Ca, 60Fe, and 107Pd. The existence of these radioactivities suggests that a nearby AGB star polluted the protosolar nebula with a small amount of neutron exposure and a low initial mass. The paper concludes by emphasizing the need for further hydrodynamical studies to address the limitations in current mixing algorithms and the role of AGB stars in galactic enrichment and solar system formation.