The problem of the indentation of an axisymmetric punch, the shape of which is simulated by a power function, into an elastic half-space when there is a liquid forming a meniscus in the gap is considered. The results are used to analyse the dependence of the contact characteristics and the capillary adhesion forces on the amount of liquid in the meniscus, the value of the applied load and the punch shape. The range of applicability of the simplified approach in which elastic deformations of the half-space outside the contact area are ignored is estimated.