A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantage of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package AUTO and provide example computations for the FitzHugh–Nagumo and the complex Ginzburg–Landau equation.