Many applications in computer vision and computer graphics require the definition of curves and surfaces. Implicit surfaces are a popular choice for this because they are smooth, can be appropriately constrained by known geometry, and require no special treatment for topology changes. In this paper we use Gaussian processes for this and derive a covariance function equivalent to the thin plate spline regularizer which has desirable properties for shape modelling. We demonstrate our approach for both 2D curves and 3D surfaces. The benefit of using a Gaussian process for this is the meaningful probabilistic representation of the function.