In Chap. 2 we explained how, for linear dynamical systems, the qualitative as well as the quantitative behaviour can be described in terms of eigenvalues and (generalized) eigenvectors. Thus, there is a systematic and feasible method for deducing the behaviour of orbits from the matrix that generates the dynamical system. For nonlinear dynamical systems, no such method exists in general. However, to characterize local dynamics near simple orbits (in particular, steady states and cycles), one can, as will be shown in detail in this chapter, rely generically on the information provided by a linear system that gives the best possible local approximation of the nonlinear system. So this chapter is all about linearization and conclusions that can be drawn concerning nonlinear systems from an analysis of linear systems obtained via linearization.