Non-Euclidean thin plates arise in different circumstances: differential growth, swelling, shrinking or plastic deformations can set the geometry of an elastic body to a preferred "target metric". In our model, the latter plays the main role in determining the shape of the plate. We use analytical techniques in the context of calculus of variations to predict the behavior of these structures for their very thin limits. We will moreover discuss a disparity between the theoretical analysis and experimental data, in which a sharp qualitative contrast between the negative and positive constant curvature cases has been observed.