The accurate approximation by low-order finite elements on uniform meshes leads to unaffordable computational complexity for important classes of PDEs. Specific tailored meshes and/or basis functions can be used in order to get more efficient discretizations. The construction of these methods typically requires specialized analytical tools. Alternatively, an effective technique to improve efficiency is the Tensor Train (TT) compression of the extremely large finite element coefficient vectors parametrizing the uniform, low-order discretization. The TT decomposition is a fairly recent tool, which has been introduced in 2011 by I. V. Oseledets to tackle the "curse of dimensionality" of tensors. The aim of this talk is to introduce the TT format for matrices and vectors, with a focus on its stability and on the complexity reduction that the TT representation permits. Then the multilevel low-rank approximation for the solution of PDEs will be addressed, which was first presented in 2018 by V. Kazeev and C. Schwab. Finally, some open problems will be discussed.