The quantum and classical transport in two-dimensional systems are investigated theoretically, with an emphasis on the scaling and the universality at the integer quantum Hall plateau transitions. The two-terminal zero-temperature conductances of two-dimensional disordered electron systems are calculated numerically with the multichannel Landauer formula by employing the transfer matrix technique within the tight-binding model. By applying the one-parameter scaling idea, all data for system sizes from 16 x 16 to very large sizes 192 x 192 and for energies within the first Landau band were scaled beautifully to one universal curve. The localization length was found to diverge at the critical point with an exponent [nu] =2.37, in accord with previous calculations and in support of the idea of universality. The critical conductance, extrapolated to macroscopic sizes through the analysis of finite size corrections from irrelevant operators, was found to be < Gc> =0.506e2[over]h, in excellent agreement with the value [sigma]xx=0.5e2[over]h predicted by an analytical theory, but in disagreement with a recent calculation based on the network model, < Gc> =0.58e2[over]h. This small but significant difference may indicate that the two-terminal critical conductance at the localization-delocalization transition is non-universal, unlike the Hall conductances which are precisely quantized due to topological invariance. Scaling of conductance is also extended to systems with anisotropic hopping integrals with the same critical exponent. However, the geometrical mean of the averaged critical conductances in the two directions does not seem to stay as a constant independent of the anisotropy, casting further doubt on the universality of the critical conductance. The statistical distribution of the conductance at the critical point showed a marked difference between the two directions in aniostropic systems. The distributions has a large weight in small G in the weak coupling direction while the distribution for the strong coupling direction shows a large dip at small G. Monte Carlo simulations on the classical adatom diffusion reveals the influence of the STM tip field on the diffusion process in the early non-equilibrium stage as seen by the change of the shape and the decay constant of the density-density correlation functions.