Spontaneous imbibition occurs in a porous medium when, driven by capillary forces, a wetting fluid such as water displaces a nonwetting fluid like oil or air. This is a common phenomenon, relevant to many processes such as groundwater contamination, steam migration in geothermal systems, carbon dioxide sequestration, and oil recovery. Statistical physics indicates how the position of the leading edge of the water scales with time. However, although the phenomenon has been studied for more than 90 years, not even simple questions like the influence of viscosities on spontaneous imbibition were well understood.