Recent studies of the nearest star-forming clouds of the Galaxy at submillimeter wavelengths with the Herschel Space Observatory have provided us with unprecedented images of the initial conditions and early phases of the star formation process. The Herschel images reveal an intricate network of filamentary structure in every interstellar cloud. These filaments all exhibit remarkably similar widths – about a tenth of a parsec – but only the densest ones contain prestellar cores, the seeds of future stars. The Herschel results favor a scenario in which interstellar filaments and prestellar cores represent two key steps in the star formation process: first turbulence stirs up the gas, giving rise to a universal web-like structure in the interstellar medium, then gravity takes over and controls the further fragmentation of filaments into prestellar cores and ultimately protostars. This scenario provides new insight into the origin of stellar masses and the star formation efficiency in the dense molecular gas of galaxies. Despite an apparent complexity, global star formation may be governed by relatively simple universal laws from filament to galactic scales.