Localized Model Order Reduction In the last decades numerical simulations based on partial differential equations (PDEs) have significantly gained importance in engineering applications. However, both the geometric complexity of the considered structures, as say ships, aircrafts, and turbines, and the intricacy of the simulated physical phenomena often make a straightforward application of say the Finite Element method prohibitive. This is particularly true if multiple simulation requests or real-time simulation response is desired as in engineering design and optimization. One way to tackle such complex problems is to exploit the natural decomposition of the structures into components and apply domain decomposition methods. In this talk we introduce local approximation spaces for some domain decomposition methods, which are optimal in the sense that they minimize the approximation error among all spaces of the same dimension. Subsequently, we generalize the procedure to parameter dependent PDEs facilitating a real-time simulation response. Moreover, we introduce a reliable and efficient a posteriori error estimator for the respective localized model order reduction procedure. The error estimator is based on local component-wise element indicators and may hence be employed within an adaptive scheme. The fact that the occurring constants in the a posteriori error estimator are localized constants associated with subdomains and not the global computational domain allows us to obtain an a posteriori error estimator, which provides a very sharp bound. We provide several numerical experiments that demonstrate an exponential convergence of the reduced approximation also for irregular geometries and show that the derived a posteriori error estimator provides a sharp and effective bound of the energy error between the reduced approximation and the reference finite element solution.