In this chapter, we want to discuss final state observability estimates. Let X be a Banach space and \((S(t))_{t\ge 0}\) an operator semigroup on X. Moreover, let Y be another Banach space and \(C\in \mathscr {L}(X,Y)\) . We call C the observation operator. We say that such a semigroup satisfies a final state observability estimatesFinal state observability estimates with respect to some Banach space Z of functions on [0, T] with values in Y, if there exists \(C_{\textrm{obs}}\ge 0\) such that \( \left\Vert S(T)x\right\Vert _X \le C_{\textrm{obs}} \left\Vert CS(\cdot )x\right\Vert _{Z} \qquad (x \in X). \) Typical applications arise from evolution equations on some function space over (a subset of) \(\mathbb {R}^d\) , where C is a restriction operator to a suitable subset \(\Omega \) of \(\mathbb {R}^d\) such that one wants to control the final state on all of \(\mathbb {R}^d\) by just measuring the evolution on the subset \(\Omega \) .