The s-process as a singular process is something of a misnomer. We know that the main s-process, producing elements up to lead, operates in AGB stars and the weak s-process, producing elements up to around A~100, operates in massive stars. Another s-process, the enhanced s-process, has been suggested as a possible explanation for the observations of some stars enriched in s elements in globular clusters, iron-poor low-mass stars in the Milky Way halo and some very metal-poor stars for which AGB stars cannot yet have contributed to the s-process. This enhanced s-process is thought to arise from rapidly rotating metal-poor massive stars which can synthesise heavier elements than usual in massive stars in the so-called "weak" s-process but can do it sooner than the cosmically tardy "main" s-process taking place in asymptotic giant branch stars. There are a number of important nuclear reactions which contribute to the yields from the enhanced s-process, notably the neutron production reaction 22Ne(alpha,n) and the competing 22Ne(alpha,gamma) reaction, and the efficacy of the neutron recycling in the competition of the 17O(alpha,gamma) and 17O(alpha,n) reactions. In this talk, I will discuss the current status of these reactions especially in the context of recently updated nuclear data from a variety of different experiments, and give some perspectives on the future requirements for studying the enhanced s process.