Recently, Fourier transform-based tensor representations have been suggested for processing and analyzing multi-dimensional data arising from real-world applications, where transform and the corresponding latent tensor serve as the two building blocks of this framework. From both perspectives, this talk will review recent progress in transform-based tensor representations. On one hand, we will discuss the evolution of the transform from linear and shallow ones to nonlinear and deep ones. On the other hand, we will discuss the evolution of the corresponding latent tensor from shallow and untrained ones to deep and pretrained ones Finally, we will also discuss the limitations and future possibilities of the transform-based tensor representation.