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 present an overview of MFEM (https://mfem.org), a library for high-order finite element methods, which powers HPC applications in a wide variety of fields. We also report on recent work in the Center for Efficient Exascale Discretizations (https://ceed.exascaleproject.org) a co-design center in the US Exascale Computing Project focused on next-generation discretization software and algorithms, where MFEM plays a major role. We will 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, high-order data analysis and visualization, and demonstrate their impact in several large-scale applications from the US Department of Energy.