Numerical linear algebra is principally concerned with solving two types of problems: A x = b and A v = lambda v. In both cases, it can be surprisingly hard to produce accurate values for solutions when inputs to the problem may be off by very small amounts. In applications, the problems being solved are often intended to approximate operators in infinite dimensional settings. The problems with the first type of calculation are well understood, and very satisfactory theorems about approximation of solutions of high (or even infinite) dimensional operators are known. In the case of the eigenvalue problem the situation is more interesting. For normal operators a great deal is known, however, for non-normal operators the situation is not as well understood. Operators may converge without spectra (eigenvalues) converging. We will explore the concept of the $\epsilon$-pseudospectrum which "repairs" some of the problems that the spectrum has, and we will look at some of the tools that are available for computing the pseudospectrum. In addition we will discuss some of the applications of these ideas to analysis of the stability of solutions of PDEs.