In this talk I will discuss algebraic approaches to solving large-scale linear systems and large-scale eigenvalue problems efficiently. The techniques, which I will discuss, have in common that they implicitly use information about the analytic model, while the approach itself is algebraic and uses only a few key parameters. The talk will give an overview of two approaches. One is based on hierarchical matrix approximation techniques. The other uses multilevel incomplete factorization. For a large class of partial differential equations, and related problems, these approaches allow us to solve linear systems of equations and eigenvalue problems easily with only minor problem-specific changes.