Within the project Dynamical Systems Uncertainty Quantification for Climate Systems (DUCS), we are going to perform uncertainty quantification of random sets that appear as a result of forward propagation of parameter uncertainty in dynamical systems. The aim is to gain a better understanding of the geometry of these random objects in phase space. Crucial observables, such as a notion of expectation of the random set, will be approached from a dynamical systems point of view. Furthermore, suitable measures to perform a risk assessment, e.g., based upon second moments of random sets, will be designed and computed. Numerical simulation studies for several conceptual climate models with an increasing degree of complexity will make the new mathematical tools available to geoscience, and also directly serve as cross-validation for the theoretical results. In summary, we will provide set-theoretic uncertainty quantification tools that go beyond current local linearisation methods and allow for a global assessment of bifurcation structures in complex systems and thereby, improve on the risk assessment of tipping.