Theoretical studies on surface electrical conduction on nanometer scales are presented. First the mechanism of the observation of surface states in scanning tunneling microscopy is discussed, where it is essential to take account of the current flowing parallel to surfaces. Second the electrical conductance in double-tip scanning tunneling microscopy is discussed. The double-tip conductance of group-IV semiconductor (111)2×1surfaces is calculated, where it is found that the buckling of π-bonded chains induces asymmetry between valence and conduction bands and reduces the anisotropy of conduction. Finally the electrical resistance of monatomic steps of the Si(111)√3×√3-Ag surface is calculated. The calculated value of conductance reproduces well the experimental values. The transmission of Bloch states is discussed using generalized phase and amplitude of waves including evanescent waves. The resistance arises from both the difference in Bloch wave numbers and the discontinuity of the logarithmic derivatives of the periodic part of Bloch waves.