Clustering in high-dimensional spaces is a difficult problem which is recurrent in many domains, for example in image analysis. The difficulty is due to the fact that high-dimensional data usually live in different low-dimensional subspaces hidden in the original space. This paper presents a family of Gaussian mixture models designed for high-dimensional data which combine the ideas of dimension reduction and parsimonious modeling. These models give rise to a clustering method based on the Expectation-Maximization algorithm which is called High-Dimensional Data Clustering (HDDC). In order to correctly fit the data, HDDC estimates the specific subspace and the intrinsic dimension of each group. Our experiments on artificial and real datasets show that HDDC outperforms existing methods for clustering high-dimensional data.