Reduced basis (RB) methods have been developed during the last decades with the aim to efficiently compute numerical solutions for parametrized applications. While the application of certified reduced basis methods spans an increasing array of problem types, the computational challenges associated with high dimensional parameter spaces remain very significant. In this presentation, we shall elaborate on some of the main bottlenecks associated with the high dimensional case and discuss a number of different ideas that allow for substantial reductions in the offline or online stages of the reduced basis methods. We consider the use of adaptive sampling in the greedy approximation. The development of ANOVA based techniques in combination with reduced basis methods to compress the parameter space, and an adaptive "hp" approach to decompose the parameter domain, allowing for a substantial acceleration of the method, both at the offline and the online stage.