Parametric model order reduction using reduced basis methods can be an effective tool
for obtaining quickly solvable reduced order models of parametrized partial differential
equation problems. With speedups that can reach several orders of magnitude, reduced
basis methods enable high fidelity real-time simulations of complex systems and dramatically
reduce the computational costs in many-query applications. In this contribution we
analyze the methodology, mainly focussing on the theoretical aspects of the approach.
In particular we discuss what is known about the convergence properties of these methods:
when they succeed and when they are bound to fail. Moreover, we highlight some recent
approaches employing nonlinear approximation techniques which aim to overcome the
current limitations of reduced basis methods.