We present sampling results for certain classes of two-dimensional signals that are not bandlimited, but have a parametric representation with a finite number of degrees of freedom. While there are many such parametric signals, it is often difficult to propose practical sampling schemes, therefore, we will concentrate on those classes for which we are able to give exact sampling algorithms and reconstruction formulas. We analyze in detail a set of 2-D Diracs and extend the results to more complex objects such as lines and polygons. Unlike most multidimensional sampling schemes, the methods we propose perfectly reconstruct such signals from a finite number of samples. Some of the techniques we use are already encountered in the context of harmonic retrieval and error correction coding. In particular, singular value decomposition based methods and the annihilating filter approach are both explored as inherent parts of the proposed algorithms. Applications of our results can be found in astronomical signal processing, image processing and in some classes of identification problems.