When the sliding of a viscous grain boundary is accommodated by grain-boundary diffusion, we evaluate the sliding rate and the stress distribution on the boundary, by employing the energy-balance method developed by Mori et al. [Mori T, Nakasone Y, Taya M, Wakashima K. Phil Mag Lett 1997;75:359 and Mori T, Koda M, Monzen R, Mura T. Acta Metall 1983;31:275]. The phenomena analyzed are the sliding of non-planar grain boundaries, the sliding of boundaries containing particles and the rotation of polygonal grains in two dimensions. For given grain-boundary configurations or grain shapes, we obtain respective analytical expressions for the sliding and rotation rates in a steady state, as a function of the viscosity for the boundary sliding. The obtained sliding and rotation rates are different from the existing ones. In particular, the sliding rates of non-planar grain boundaries have different expressions from the classical predictions by Raj and Ashby [Raj R, Ashby MF. Metall Trans 1971;2:1113]. The sliding/rotation rate and the normal stress on the grain boundary decrease with an increase in the boundary viscosity. As the boundary viscosity becomes sufficiently large, the deformation rate is dominated by viscous grain-boundary sliding, and the contribution of grain-boundary diffusion becomes negligible.