Optimal Algorithms for Eigenvalue Problems and the SVD

What is the optimal algorithm for computing the eigenvalues/eigenvectors of an arbitrary matrix? What about the singular value decomposition (SVD)? What does optimal even mean in this context?  I'll survey some recent efforts to answer these questions, from both a theoretical computer science and classical numerical analysis perspective. I'll also discuss related open research directions; despite the fact that the earliest eigenvalue algorithms date back nearly 200 years, there's still lots to be discovered here!