This paper addresses the problem of describing surfaces using local features and descriptors. While methods for the detection of interest points in images and their description based on local image features are very well understood, their extension to discrete manifolds has not been well investigated. We provide a methodological framework for analyzing real-valued functions defined over a 2D manifold, embedded in the 3D Euclidean space, e.g., photometric information, local curvature, etc. Our work is motivated by recent advancements in multiple-camera reconstruction and image-based rendering of 3D objects: there is a growing need for describing object surfaces, matching two surfaces, or tracking them over time. Considering polygonal meshes, we propose a new methodological framework for the scale-space representations of scalar functions defined over such meshes. We propose a local feature detector (MeshDOG) and region descriptor (MeshHOG). Unlike the standard image features, the proposed surface features capture both the local geometry of the underlying manifold and the scale-space differential properties of the real-valued function itself. We provide a thorough experimental evaluation. The repeatability of the feature detector and the robustness of feature descriptor are tested, by applying a large number of deformations to the manifold or to the scalar function.