Cellular membranes bend and curve into a multitude of shapes as they perform various functions. These deformations make use of the remarkable material properties of biological membranes inherent in their nature as two-dimensional fluids. The curvature of membranes is controlled by the constituent proteins and lipids, but conversely, curvature itself provides mechanisms for organizing mobile membrane molecules. In this article we survey recent experiments that have uncovered intriguing connections between mechanics and biochemistry at membranes, focusing on the influence of molecular shape on curvature, links between phase separation and curvature, and membrane bending at inter-cellular contacts. We describe the concepts that emerge from these studies, especially the existence of long-range, curvature-mediated mechanisms for spatial organization in membranes, and highlight open areas for future research.