The methodology of unfitted finite element methods, i.e. methods that are able to cope with interfaces or boundaries that are not aligned with the grid, have been investigated for different problems in recent years, in particular for spatial problems. The benefit of flexible handling of geometries becomes especially interesting with time-depending problems on moving domains. Hence, we investigate generalisations of well-established techniques from the spatial case to the moving domain case.[1] For instance, the isoparametric mesh curvature of [2] is generalised to a space-time setting, yielding a higher order accurate discrete geometry in space and time. This gives a solution to the issue of numerical integration on implicit domains, which is one of the key challenges for unfitted methods. Another one is the stability under ill-posed cut configurations. To this end, we present a space-time version of a Ghost penalty stabilisation, which we call the direct Ghost penalty. To apply these problem-independent techniques to an example discretisation, we focus on a convection-diffusion problem and design a discontinuous in time, conforming in space Galerkin discretisation. The higher order convergence property of this method is investigated numerically and we briefly mention results from numerical analysis.[3] For computational benefits, we also introduce a continuous in time Galerkin variant. Apart from the bulk convection-diffusion problem, we also investigate a coupled convection-diffusion problem on bulk and surface. [1] Fabian Heimann, Christoph Lehrenfeld, Janosch Preuß. Geometrically higher order unfitted space-time methods for PDEs on moving domains. SIAM Journal on Scientific Computing, 45 (2) B139-165. URL: https://epubs.siam.org/doi/full/10.1137/22M1476034 [2] Christoph Lehrenfeld. High order unfitted finite element methods on level set domains using isoparametric mappings. Computer Methods in Applied Mechanics and Engineering, 300:716 – 733, 2016. URL: http://www.sciencedirect.com/science/article/pii/S0045782515004004. [3] Fabian Heimann, Christoph Lehrenfeld. Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains: Geometry Error Analysis. arXiv:2311.02348. URL: https://arxiv.org/abs/2311.02348