The adhesive contact problems of a conical frustum punch indenting a transversely isotropic elastic half space and an orthotropic elastic half space are analytically studied in this thesis work. To solve the problem involving a transversely isotropic half space, the harmonic potential function method and the Hankel transform are employed, which lead to a general closed-form solution for the adhesive contact problem. For the case with an orthotropic half space, the problem of a point load applied on the half space is first solved by using the double Fourier transform method. The solution for the adhesive contact problem is then obtained through integrating the former solutions over the punch surface.