The general theory of axisymmetric hardness tests on nonlinear media is approached from the standpoint of similarity transformations. It is shown how an entire process of indentation can be made to depend on the solution of just one boundary-value problem in scaled variables and with a fixed geometry. Once this single auxiliary solution has been obtained, the values of all physical quantities in the original problem can be generated readily at any stage without further numerical error. Even by themselves the similarity relations provide valuable information about (for example) an invariant connection between the depth of penetration and the radius of contact, or about the variation of penetration with time in a creep test under dead load. Two kinds of material behaviour are considered: (a) nonlinear elastic (modelling strain-hardening plasticity) and (b) nonlinear viscous (modelling secondary creep). In either category the constitutive specification is sufficiently flexible to represent a wide range of actual responses in the context of hardness testing. The analysis for case (a) extends a theory of ball indentation by Hill et al. to a class of indenters with shapes varying from flat to conical. It also prepares the ground for case (b) which is more difficult and calls for a quite different auxiliary problem.