  We give an overview of some recent results on operator-valued $(L^{p},L^{q})$
Fourier multipliers and stability theory for evolution equations. The aim is to
provide a relatively nontechnical introduction to the underlying ideas,
emphasizing the connection between the two areas. We also indicate how
operator-valued $(L^{p},L^{q})$ Fourier multipliers can be applied to
functional calculus theory.
