Unfitted finite element methods are interesting for applications in fluid dynamic simulations because of their ability to cope with complicated geometries by means of an unaligned background mesh. The task of deriving numerical quadrature rules within these methods is particularly straightforward for geometry approximations which are elementwise linear or of second order in relevant norms. However, higher order finite element simulations can provide a desirable error scaling behaviour. Hence, we want to present a way to obtain unfitted higher order finite element simulations, in particular for a convection-diffusion problem on a moving domain. [1] We focus on a Discontinuous Galerkin in time space-time method, which comes in a very general formulation in regards to the discretisation order. Concerning the spatial discretisation, we generalise an isoparametric mapping known from merely spatial problems. [2] We briefly mention results from numerical analysis yielding inf-sup stability and optimal order convergence in reasonable norms, under certain assumptions. [3,4] Moreover, we present numerical results confirming these higher order convergence properties in two and three spatial dimensions. References: [1] Fabian Heimann, Christoph Lehrenfeld, Janosch Preuß. Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains. Arxiv pre-print, https://arxiv.org/abs/2202.02216. [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. [3] Janosch Preuß. Higher order unfitted isoparametric space-time fem on moving domains. Master’s thesis, University of Göttingen, 2018. URL: http://cpde.math.uni-goettingen.de/data/Pre18_Ma.pdf. [4] Fabian Heimann. On discontinuous- and continuous-in-time unfitted space-time methods for pdes on moving domains. Master’s thesis, University of Göttingen, 2020. URL: https://cpde.math.uni-goettingen.de/data/Hei20_Ma.pdf.