Brazing and soldering, as advanced manufacturing processes, are of significant importance to industrial applications. It is widely accepted that joining by brazing or soldering is possible if a liquid metal wets the solids to be joined. Wetting, hence spreading and capillary action of liquid metal (often called filler) is of significant importance. Good wetting is required to distribute liquid metal over/between the substrate materials for a successful bonding. Topographically altered surfaces have been used to exploit novel wetting phenomena and associated capillary actions, such as imbibitions (a penetration of a liquid front over/through a rough, patterned surface). Modification of surface roughness may be considered as a venue to tune and control the spreading behavior of the liquids. Modeling of spreading of liquids on rough surface, in particular liquid metals is to a large extent unexplored and constitutes a cutting edge research topic. In this dissertation the imbibitions of liquid metal has been considered as pertained to the metal bonding processes involving brazing and soldering fillers. First, a detailed review of fundamentals and the recent progress in studies of non-reactive and reactive wetting/capillary phenomena has been provided. An imbibition phenomenon has been experimentally achieved for organic liquids and molten metals during spreading over topographically modified intermetallic surfaces. It is demonstrated that the kinetics of such an imbibition over rough surfaces follows the Washburn-type law during the main spreading stage. The Washburn-type theoretical modeling framework has been established for both isotropic and anisotropic non-reactive imbibition of liquid systems over rough surfaces. The rough surface domain is considered as a porous-like medium and the associated surface topographical features have been characterized either theoretically or experimentally through corresponding permeability, porosity and tortuosity. Phenomenological records and empirical data have been utilized to verify the constructed model. The agreement between predictions and empirical evidence appears to be good. Moreover, a reactive wetting in a high temperature brazing process has been studied for both polished and rough surfaces. A linear relation between the propagating triple line and the time has been established, with spreading dominated by a strong chemical reaction.