Space Time Adaptive Processing (STAP) is a two-dimensional adaptive filtering technique which uses jointly temporal and spatial dimensions to suppress disturbance and to improve target detection. Disturbance contains both the clutter arriving from signal backscattering of the ground and the thermal noise resulting from the sensors noise. In practical cases, the STAP clutter can be considered to have a low rank structure. Using this assumption, a low rank vector STAP filter is derived based on the projector onto the clutter subspace. With new STAP applications like MIMO STAP or polarimetric STAP, the generalization of the classic filters to multidimensional configurations arises. A possible solution consists in keeping the multidimensional structure and in extending the classic filters with multilinear algebra. Using the low-rank structure of the clutter, we propose in this paper a new low-rank tensor STAP filter based on a generalization of the Higher Order Singular Value Decomposition (HOSVD) in order to use at the same time the simple (for example time, spatial, polarimetric, …) and the combined information (for example spatio-temporal). Results are shown for two cases: classic 2D STAP and 3D polarimetric STAP. In the classic case, vector and tensor filters are equivalent. In the polarimetric case, we show the enhancement of the tensor filter.