It is well known that the real parts of the eigenvalues of the coefficient matrix in an autonomous linear differential equation determine the stability properties of solutions. However, for nonautonomous linear differential equations simple examples show that the eigenvalues of the coefficient matrix function can give incorrect stability information. We consider different definitions of spectrum for nonautonomous linear differential equations and their uses. We review known perturbation results, derive some consequences, and show relationships between different types of spectrum with

an eye toward the impact on numerical approximation of spectral intervals.

Joint work with Luca Dieci.