The finite-element method is used to model the elastic-plastic indentation response of a flat extensive specimen for the case of a spherical indenter. The work highlights several interesting finite-element modelling techniques and provides insight into the physical processes involved in elastic-plastic indentation of certain structural ceramics. Full details of the stress distribution are given and compared with the results of elastic formulae. This work has particular application to the modelling of physical phenomena of deformation in ceramic materials in machining, wear, bearings and hardness testing.