As optical fiber communications and fiber lasers approach fundamental limits, interest in multimode fibers has grown. Optical fibers designed to support multiple transverse modes offer opportunities to study wave propagation in a setting that is intermediate between single-mode fiber and free-space propagation. A variety of qualitatively-new phenomena have been observed recently in multimode fibers. By varying the launched spatial modes, it is possible to generate megawatt ultrashort pulses tunable between 1550 and 2200 nm, dispersive waves over one octave in frequency, intense combs of visible light, or continua that span multiple octaves. New instabilities, which are spatiotemporal in nature, occur. A few of these new phenomena will be presented along with their connection to multimode soliton dynamics. Possible directions for studies of new nonlinear wave physics in multimode fibers will be discussed along with potential applications.