Multilevel block Toeplitz matrices arise in many applications, for instance when higher-order discretizations are used for scalar PDEs or systems of PDEs are to be solved. For large-scale problems multigrid methods are often the method of choice, as they provide an efficient way of solving the associated linear systems. The analysis of multigrid methods for structured matrices, i.e., Toeplitz matrices or circulant matrices, and traditional multilevel theory is an established technqiue. For scalar problems, including those arising from the discretization of PDEs, it has been studied intensively. Recently, we started transfering these results to the systems case that results in block-Toeplitz matrices or block-circulant matrices [1]. Besides studying higher-order discretizations of scalar PDEs, certain systems of PDEs also fit in this framework. Systems of PDEs that lead to saddle point structure, like the Stokes equations, need another approach. Based on a result by Notay [3] we were able to establish convergence for these matrices, as well [2]. In the talk the analysis technique, the derived sufficient conditions for optimal convergence and numerical results will be presented. [1] M. Bolten, M. Donatelli, P. Ferrari, and I. Furci. A symbol based analysis for multigrid methods for block-circulant and block-Toeplitz systems. SIAM J. Matrix Anal. Appl., 43(1):405–438, 2022. [2] M. Bolten, M. Donatelli, I. Ferrari, and I. Furci. Symbol based convergence analysis in multigrid methods for saddle point problems. Linear Algebra Appl., 671:67–108, 2023. [3] Y. Notay. A new algebraic multigrid approach for Stokes problems. Numer. Math., 132(1):51–84, 2016.