Exascale architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially-assembled high-order finite element methods, since these methods can increase the accuracy and lower the computational time due to reduced data motion. In this talk we report on recent work in MFEM [1], a library for high-order finite element methods, which powers HPC applications in a wide variety of fields [2]. We describe recent MFEM advancements in performance optimizations for GPU architectures, high-order finite element benchmarks and miniapps, scalable unstructured adaptive mesh refinement, matrix-free preconditioning for partially assembled operators, many of which started under the Center for Efficient Exascale Discretizations [3], a co-design center in the US Exascale Computing Project. We demonstrate the impact of these developments in several large-scale HPC simulations, including the 2025 Gordon Bell winner [4]. [1] MFEM Library, https://mfem.org. [2] J. Andrej, N. Atallah, J.-P. Backer, J.-S. Camier, D. Copeland, V. Dobrev, Y. Dudouit, T. Duswald, B. Keith, D. Kim, T. Kolev, B. Lazarov, K. Mittal, W. Pazner, S. Petrides, S. Shiraiwa, M. Stowell, and V. Tomov, High-performance finite elements with MFEM, Int. J. High Perform. Comput. Appl., 38, pp. 447–467, 2024, https://doi.org/10.1177/10943420241261981. [3] Center for Efficient Exascale Discretizations, https://ceed.exascaleproject.org. [4] S. Henneking, S. Venkat, V. Dobrev, J. Camier, T. Kolev, M. Fernando, A. Gabriel, O. Ghattas, Real-time Bayesian inference at extreme scale: A digital twin for tsunami early warning applied to the Cascadia subduction zone, SC25 proceedings, pp. 60–71, 2025, https://doi.org/10.1145/3712285.3771787.