Microwave air breakdown at dielectric surface is investigated by numerically solving the fluid-based plasma equations coupled with the Maxwell equations. The plasma formation and microwave scattering and absorption by plasma are investigated by one-dimensional (1D) and two-dimensional (2D) models. In the 1D model, it is found that at the initial stage of microwave breakdown, the plasma develops in the whole plasma region. As time increases, the plasma in the upstream grows much faster than in the downstream. Although the electron density distributions for ne = 0 and j = 0 are different, the microwave reflection, absorption and transmission are almost the same. It is found that the electron number density in the upstream region for 20 mm is larger than for 5 mm. In the 2D model, it is found for TE10 mode that the plasmoid first grows in the middle of waveguide until its density becomes large enough to diffract the incident field, then the plasma region moves along the surface to both sides. The plasma region cannot reach the wall of waveguide, where the electric field is smaller than the breakdown threshold. After comparison between the computational and experimental results, it is found that the simulated absorbed power is larger than the measured one, and the transmitted power is smaller than than measured one. The reason is that the initial electron densities in 1D and 2D simulation are both assumed to cover the whole dielectric surface, but the plasma in experiment develops in a very small region.