Vector and tensor fields are multidimensional data, where in each pixel/voxel the field is assigned to a vector or a matrix. The fields describe particle velocity, optical flow, stress and conductivity tensors, and similar phenomena. One of the challenging tasks is the invariant detection of patterns of interest. Invariants to total rotation and affine transformation of the field are desirable to accomplish this task. In this chapter, we review a recent development of this research area and show several practical applications on real data.