The recognition that physical space (or space-time) is curved is a product of the general theory of relativity, such as dramatically shown by the 1919 solar eclipse measurements. However, the mathematical possibility of non-Euclidean geometries was recognized by Gauss more than a century earlier, and during the nineteenth century mathematicians developed the pioneering ideas of Gauss, Lobachevsky, Bolyai and Riemann into an elaborate branch of generalized geometry. Did the unimaginative physicists and astronomers ignore the new geometries? Were they considered to be of mathematical interest only until Einstein entered the scene? This paper examines in detail the attempts in the period from about 1830 to 1910 to establish links between non-Euclidean geometry and the physical and astronomical sciences, including attempts to find observational evidence for curved space. Although there were but few contributors to "non-Euclidean astronomy," there were more than usually supposed. The paper looks in particular on a work of 1872 in which the Leipzig physicist K. F. Zoellner argued that the universe is closed in accordance with Riemann's geometry.