This paper reports a computational study of the indentation of a flat punch into a compressible elastic layer (with Poisson's ratio varying from 0 to 0.49) bonded to a rigid substrate. Based on the computational results and using Sneddon's solution [Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 1965;3:47] and the asymptotic solution [Jaffar MJ. A general solution to the axisymmetric frictional contact problem of a thin bonded elastic layer. Proc Inst Mech Eng C 1997;211:549; Yang FQ. Asymptotic solution to axisymmetric indentation of a compressible elastic thin film. Thin Solid Films 2006;515:2274] as the two limits, a simple expression of the load-depth curve valid for an arbitrary ratio of the indenter radius to the thickness of the layer is obtained. Further, a correlation between indentation load and depth for a rigid flat punch indenting into linearly viscoelastic layers bonded to a rigid substrate is proposed by using the correspondence principle. Several procedures are suggested based on the results reported in this study to determine the viscoelastic properties of the layer in the time or frequency domains. The findings are verified by numerical examples. The results may facilitate the use of depth-sensing indentation tests to characterize the mechanical properties of polymeric films or functional coatings on hard substrates, and some biological materials, e.g. articular cartilage.