Shell elements derived from the Kirchhoff-Love theory have experienced a renaissance in the Finite Element community since the advent of isogeometric analysis. Despite numerous promising advancements in the field, several questions need to be addressed to establish isogeometric Kirchhoff-Love shells as an industry standard. In particular, it is known that the direct simulation of trimmed multi-patch shell models requires particular care, both from the point of view of the geometry as well as from the analysis side. This thesis focuses on the latter. Specifically, our goal is to develop accurate and robust algorithms for the analysis of trimmed surfaces. To achieve this, we address several important aspects: (i) we systematically study the beneficial effect of local refinement for the proper resolution of localized features of the geometry/solution on complex trimming patterns, (ii) we devise a novel a-posteriori error estimator tailored to Kirchhoff plates and Kirchhoff-Love shells which allows us to develop a fully adaptive computational framework, (iii) we present a locking- and parameter-free coupling strategy adapted from the penalty method for achieving the required C^1-continuity across patches and (iv) we provide a preliminary study on the impact of trimming on the critical time step in the scope of explicit Kirchhoff-Love shell dynamics. We numerically verify and assess the performance of the aforementioned methods on an extensive selection of benchmarks defined on trimmed domains. We systematically observe an increase in accuracy compared to other established approaches. Specifically, our methods are constructed to attain the optimal accuracy of splines. To conclude, we test the capabilities of our computational framework on several engineering structures by performing a static shell analysis on, e.g. the B-pillar of a car and the blade of a wind turbine.