In this chapter we will give an insight into modern sparse elimination methods. These are driven by a preprocessing phase based on combinatorial algorithms which improve diagonal dominance, reduce fill-in, and improve concurrency to allow for parallel treatment. Moreover, these methods detect dense submatrices which can be handled by dense matrix kernels based on multithreaded level-3 BLAS. We will demonstrate for problems arising from circuit simulation, how the improvements in recent years have advanced direct solution methods significantly.