A traditional view on solitons in optical fibers as robust particlelike structures suited for information transmission has been significantly altered and broadened over the past decade when solitons have been found to play the major role in generation of octave broad supercontinuum spectra in photonic crystal and other types of optical fibers. This remarkable spectral broadening is achieved through complex processes of dispersive radiation being scattered from, emitted, and transformed by solitons. Thus solitons have emerged as the major players in nonlinear frequency conversion in optical fibers. Unexpected analogies of these processes have been found with dynamics of ultracold atoms and ocean waves. This Colloquium focuses on recent understanding and new insights into physics of soliton-radiation interaction and supercontinuum generation.