Cells have evolved sophisticated molecular machinery and signaling pathways to sense and respond to their environment. In particular, membrane receptor interactions with ligands immobilized on structures in the cells' milieu allow them to sense the geometry of their surroundings, including the presence of surfaces or boundaries. In vivo, cells interact with curved surfaces with radii of curvature ranging from the size of a cell (50-100 microns) to millimeters; examples occur throughout the body in the form of vasculature, glands, and villi. However, remarkably little is known about cell interaction with non-planar boundaries; this dissertation probes how surface curvature influences cell alignment, internal microstructure, and migration. In the first study, mouse embryonic fibroblasts (MEFs) were cultured on glass cylinders with radii ranging from 40-200 microm. Isolated cells aligned along the axial direction on small cylinders but not large cylinders. Two distinct populations of actin stress fibers (SFs) aligned along the principal directions of the cylindrical surface: Long apical SFs spanned the nucleus and aligned along the axial direction whereas short, basal SFs beneath the nucleus aligned in the circumferential direction on small cylinders. The second study explored how these SF populations align on surfaces with non-zero Gaussian curvature. For this study, a sphere-with-skirt (SWS) surface was designed that seamlessly connects a positive Gaussian curvature spherical cap to a negative Gaussian curvature skirt. On the skirt, the SF populations again aligned along the principal directions: Apical SFs remained straight by forming chords over the concave gap and basal SFs bent along the convex direction. MEFs avoided the positive Gaussian curvature cap and instead migrated around the SWS feature in the azimuthal direction. Finally, the importance of edges between ligand-presenting and ligand-free domains was studied to discern the impact on cell alignment and migration. Edges enhanced the migration of isolated MEFs and acted as a slip boundary for cells in confluent, nematic monolayers. Topological defects emerged in corners where two edges met. These studies revealed that geometric cues in the form of surface curvature and bounding edges dictate cell alignment, cytoskeletal organization, and migration.