Upcoming exascale architectures require rethinking of the numerical algorithms used in large-scale PDE-based applications. These architectures favor algorithms, such as high-order finite elements, that expose fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. In this talk we present an overview of MFEM [1], a scalable library for high-order finite element discretization of PDEs on general unstructured grids. We also report on recent work in the Center for Efficient Exascale Discretizations [2], a co-design center in the US Exascale Computing Project focused on next-generation discretization software and algorithms. We will describe recent research on performance optimizations for GPU architectures, scalable unstructured adaptive mesh refinement and matrix-free preconditioning, with special attention to the needs for time integration methods in high-order applications. [1] MFEM: Modular finite element library, http://mfem.org. [2] Center for Efficient Exascale Discretizations, http://ceed.exascaleproject.org.