Feature selection is of great importance in data mining tasks, especially for exploring high dimensional data. Laplacian Score, a recently proposed feature selection method, makes use of local manifold structure of samples to select features and achieves good performance. However, it ignores the global structure of samples and the selected features are of high redundancy. To address these issues, we propose a feature selection method based on local and global structure preserving, LGFS in short. LGFS first uses two graphs, nearest neighborhood graph and farthest neighborhood graph to describe the underlying local and global structure of samples, respectively. It then defines a criterion to prefer the features which have good ability on local and global structure preserving. To remove redundancy among the selected features, Extended LGFS (E-LGFS) is introduced by taking advantage of normalized mutual information to measure the dependency between a pair of features. We conduct extensive experiments on two artificial data sets, six UCI data sets and two public available face databases to evaluate LGFS and E-LGFS. The experimental results show our methods can achieve higher accuracies than other unsupervised comparing methods.