In this paper, we first briefly reintroduce the 1D and 2D forms of the classical Principal Component Analysis (PCA). Then, the PCA technique is further developed and extended to an arbitrary n-dimensional space. Analogous to 1D- and 2D-PCA, the new nD-PCA is applied directly to n-order tensors (n . 3) rather than 1-order tensors (1D vectors) and 2-order tensors (2D matrices). In order to avoid the difficulties faced by tensors computations (such as the multiplication, general transpose and Hermitian symmetry of tensors), our proposed nD-PCA algorithm has to exploit a newly proposed Higher-Order Singular Value Decomposition (HO-SVD). To evaluate the validity and performance of nD-PCA, a series of experiments are performed on the FRGC 3D scan facial database.