We aim at providing a review of the state of the art of the combination of (very) high order methods and immersed/embedded boundaries. High order methods are known for providing great potential in terms of simulations cost reduction at fixed resolution. This is especially true when some error estimation strategy is available to fuel advanced h-/p-adaptive techniques [1]. On the other hand, immersed and embedded approaches are usually preferred because of their extreme geometrical flexibility, allowing to accommodate complex (and possibly moving) boundaries/interfaces with no overheads related to meshing complex curved domains, and still allowing the deployment of adaptive techniques [2,3]. Recently, several approaches providing genuinely high order generalization of immersed and embedded methods have been proposed (see [4,5,6] among others). Our focus here is on the following aspects allowing to push these methods further: • Consistency and error estimates to go beyond second order of accuracy in curved (and moving) domains; • Stability estimates: as a fundamental design criterion to increase the accuracy with no impact on robustness ; • h-/p- adaptation involving rigorous error estimates and possibly higher order of accuracy (three and beyond). The MS will collect talks focused on the design and analysis of very high order unfitted methods (consistency, stability, error estimates, adaptivity) for elliptic, parabolic, and hyperbolic PDEs, and talks focusing on the benefits of these techniques for complex flows.