ABSTRACT: With the foreseeable advent of exascale super computers, adapting algorithms and software to the new hybrid-parallel infrastructures becomes inevitable. The ESSEX project (Equipping Sparse Solvers for Exascale), a part of the Priority Programme "Software for Exascale Computing" (SPPEXA), develops and investigates programming concepts and numerical algorithms for scalable, efficient and robust iterative sparse matrix applications on exascale systems. Within the project, the group of Bruno Lang from the University of Wuppertal, Germany, focuses on projection based iterative eigensolvers to tackle large scale sparse Hermitian eigenproblems in standard and generalized form, using an expanded implementation of the FEAST method and Chebyshev filter diagonalization (ChebFD) that adds improvements and provides new features, named BEAST. We provide an introduction to some of the fundamental ideas behind the main features of the BEAST eigensolver.