A finite-difference formulation of the coaxial-loop or wire-loop transient electromagnetic (EM) prospecting systems is used to model the fields from a buried cylindrical conductor whose axis is coincident with that of the field system. Solutions are obtained directly in the time domain. The formulation is implicit and two-dimensional (2-D) in space. The variable-directions method reduces each advance of one step in time from one 2-D problem to a large number of one-dimensional (1-D) problems. The result is a reduction in computational effort. To avoid including the air in the finite difference grid, an integral equation is used to formulate the surface boundary condition. Thus, two sets of 1-D finite difference solutions and Fredholm integral equation solution are required for each step forward in time. Comparison with analytical solutions shows excellent agreement in the case of a four layer earth. Inversion of transients to a stratified model can be useful if the effect of finite conductor size is taken into account. For cylindrical conductors with lateral extent comparable to or larger than the source-receiver separation, the inversion results are valid. For conductors with lateral extent small compared with source-receiver separation, the inversion wi;; yield a stratified model which shows better agreement between actual and inverted thicknesses than resistivities.--Modified journal abstract.