The extraordinary responsiveness and large diversity of self-assembled structures of liquid crystals are well documented and they have been extensively used in devices like displays. For long, this application route strongly influenced academic research, which frequently focused on the performance of liquid crystals in display-like geometries, typically between flat, rigid substrates of glass or similar solids. Today a new trend is clearly visible, where liquid crystals confined within curved, often soft and flexible, interfaces are in focus. Innovation in microfluidic technology has opened for high-throughput production of liquid crystal droplets or shells with exquisite monodispersity, and modern characterization methods allow detailed analysis of complex director arrangements. The introduction of electrospinning in liquid crystal research has enabled encapsulation in optically transparent polymeric cylinders with very small radius, allowing studies of confinement effects that were not easily accessible before. It also opened the prospect of functionalizing textile fibers with liquid crystals in the core, triggering activities that target wearable devices with true textile form factor for seamless integration in clothing. Together, these developments have brought issues center stage that might previously have been considered esoteric, like the interaction of topological defects on spherical surfaces, saddle-splay curvature-induced spontaneous chiral symmetry breaking, or the non-trivial shape changes of curved liquid crystal elastomers with non-uniform director fields that undergo a phase transition to an isotropic state. The new research thrusts are motivated equally by the intriguing soft matter physics showcased by liquid crystals in these unconventional geometries, and by the many novel application opportunities that arise when we can reproducibly manufacture these systems on a commercial scale. This review attempts to summarize the current understanding of liquid crystals in spherical and cylindrical geometry, the state of the art of producing such samples, as well as the perspectives for innovative applications that have been put forward.