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Coronal mass ejections (CMEs) may disturb the solar wind either by overtaking it, or by expanding into it, or both. CMEs whose front moves faster in the solar wind frame than the fast magnetosonic speed, drive shocks. Such shocks are important contributors to space weather, by triggering substorms, compressing the magnetosphere and accelerating particles. In general, near 1 AU, CMEs with speed greater than about 500 km s$^{-1}$ drive shocks, whereas slower CMEs do not. However, CMEs as slow as 350 km s$^{-1}$ may sometimes, although rarely, drive shocks. Here, we study these slow CMEs with shocks and investigate the importance of CME expansion in contributing to their ability to drive shocks and in enhancing shock strength. Our focus is on CMEs with average speeds under 375 km s$^{-1}$. From Wind measurements from 1996 to 2016, we find 22 cases of such shock-driving slow CMEs, and, for about half of them (11 out of the 22), the existence of the shock appears to be strongly related to CME expansion. We also investigate the proportion of all CMEs with speeds under 500 km s$^{-1}$ with and without shocks in solar cycles 23 and 24, depending on their speed. We find no systematic difference, as might have been expected on the basis of the lower solar wind and Alfv\'en speeds reported for solar cycle 24 vs. 23. The slower expansion speed of CMEs in solar cycle 24 might be an explanation for this lack of increased frequency of shocks, but further studies are required., Comment: 15 pages, 4 Figures, 5 Tables, accepted to ApJ