to the optimization problem. This permits more accurate results in terms of approximation of the optimal solution;

-   we provide efficiently computable a posteriori error estimates for all reduced quantities for different choices of the cost functional and its (approximate) gradient;

-   we rigorously prove the convergence of the TR-RB method with bilateral inequality constraints on the parameters;

-   we devise several new adaptive enrichment strategies for the progressive construction of the Reduced Basis spaces;

-   we demonstrate in numerical experiments that our new TR-RB methods outperform existing model reduction approaches for large scale optimization problems in well defined benchmark problems.

**Organization of the article.** In Section \[sec:problem\] we introduce the PDE constrained optimization problem and state first- and second-order optimality conditions. These serve as a basis for the full order discretization derived in Section \[sec:mor\]. Moreover, in Section \[sec:mor\] we introduce different strategies of model order reduction for the full order model and derive rigorous a posteriori error estimates for all equations, functionals, and gradient information. Section \[sec:TRRB\_and\_adaptiveenrichment\] is devoted to the derivation of Trust-Region – Reduced Basis methods and the presentation of the convergence analysis of the adaptive TR-RB algorithm. In addition, we discuss in detail several variants 