Various aspects of the problem of current transfer in high- superconductors (HTSs) are reviewed. The spatial inhomogeneities of various types are identified as a primary cause of non-uniformity of both normal currents and supercurrents in real samples of HTSs. The role these inhomogeneities play in transport features of the samples is discussed. The case of grain boundaries in polycrystalline samples is elaborated in detail. The local structural and transport properties of isolated grain boundaries are first reviewed and then integrated into the knowledge of global (macroscopic) charge transport. The paper emphasizes the common ingredients characterizing the transport in various forms and families of HTS samples in small magnetic fields. The phenomenon of percolation is identified as the most obvious one and is shown to dominate a large number of observations covered by this report. The experimental results focused on by this report elaborate primarily the problems of critical currents, initial dissipation and current-voltage characteristics, penetration depth, resistive and metal-insulator transition, resistance noise and magneto-optical studies of current paths. Various models for current transfer (disordered bonds, brick wall and railway switch) are also reviewed and discussed.