Wicking within woven screens has attracted considerable attention due to its important role in applications concerning phase-change heat transfer and phase separation. In the present study, horizontal spreading experiments are conducted to investigate the wicking performance of woven screens by measuring the volumetric liquid intake into the screens and the liquid propagation fronts through two perpendicular high-speed cameras. Woven screens with micro (single- and multilayer)- and nano (plain, etched, and fluoridated)-porous structures are manipulated through diffusion bonding and chemical processes. The macroscopic observation indicates the substantial enhancement of the wicking capability in multilayer structures, where the interlayer microchannels could compensate for the essential deficiency of single-layer screens by providing low-resistance flow passages. Wicking capability of water is enhanced by the hydrophilic nanograsses along the wires. Furthermore, flow mechanisms within the screens are analyzed by comparisons between apparent and saturated wicking distances. In multilayer structures, the liquid spreads along the entire cross-sectional area in etched screens, while it spreads primarily along the interlayer microchannels in plain and fluoridated screens. The influence of various fluids on the wicking behavior within the woven screens is found to be fully represented by a unique parameter that captures the effects of surface tension and dynamic viscosity in the radial flow model. This work deepens the understanding of the capillary-driven flow within the woven screens with hybrid micro-/nanoporous structures and provides guidance for the design and manufacture of highly efficient wicking structures.