Graph and hypergraph matching are important problems in computer vision. They are successfully used in many applications requiring 2D or 3D feature matching, such as 3D reconstruction and object recognition. While graph matching is limited to using pairwise relationships, hypergraph matching permits the use of relationships between sets of features of any order. Consequently, it carries the promise to make matching more robust to changes in scale, deformations and outliers. In this paper we make two contributions. First, we present a first semi-supervised algorithm for learning the parameters that control the hypergraph matching model and demonstrate experimentally that it significantly improves the performance of current state-of-the-art methods. Second, we propose a novel efficient hypergraph matching algorithm, which outperforms the state-of-the-art, and, when used in combination with other higher-order matching algorithms, it consistently improves their performance.