We investigate the cosmological implications of f(Q) gravity, which is a modified theory of gravity based on non-metricity, in non-flat geometry. We perform a detailed dynamical-system analysis keeping the f(Q) function completely arbitrary. As we show, the cosmological scenario admits a dark-matter dominated point, as well as a dark-energy dominated de Sitter solution which can attract the Universe at late times. However, the main result of the present work is that there are additional critical points which exist solely due to curvature. In particular, we find that there are curvature-dominated accelerating points which are unstable and thus can describe the inflationary epoch. Additionally, there is a point in which the dark-matter and dark-energy density parameters are both between zero and one, and thus it can alleviate the coincidence problem. Finally, there is a saddle point which is completely dominated by curvature. In order to provide a specific example, we apply our general analysis to the power-law case, showing that we can obtain the thermal history of the Universe, in which the curvature density parameter may exhibit a peak at intermediate times. These features, alongside possible indications that non-zero curvature could alleviate the cosmological tensions, may serve as advantages for f(Q) gravity in non-flat geometry.