AbstractIn this chapter we study the exponential stability of evolutionary equations. Roughly speaking, exponential stability of a well-posed evolutionary equation $$\displaystyle \left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )U=F $$ ∂ t , ν M ( ∂ t , ν ) + A U = F means that exponentially decaying right-hand sides F lead to exponentially decaying solutions U. The main problem in defining the notion of exponential decay for a solution of an evolutionary equation is the lack of continuity with respect to time, so a pointwise definition would not make sense in this framework. Instead, we will use our exponentially weighted spaces $$L_{2,\nu }(\mathbb {R};H)$$ L 2 , ν ( ℝ ; H ) , but this time for negative ν, and define the exponential stability by the invariance of these spaces under the solution operator associated with the evolutionary equation under consideration.