This report is dedicated to the statistical analysis of the space of multivariate normal probability density functions. It relies on the differential geometrical properties of the underlying parameter space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on non-linear spaces. We will first proceed to the state of the art in section 1, while expressing some quantities related to the structure of the manifold of interest, and then focus on the derivation of closed-form expressions for the mean, covariance matrix, modes of variation and normal law between multivariate normal distributions in section 2. We will also address the derivation of accurate and efficient numerical schemes to estimate the proposed quantities. A major application of the present work is the statistical analysis of diffusion tensor Magnetic Resonance Imaging. We show promising results on synthetic and real data in section 3