In this presentation we will present recent advances in the numerical approximation of PDEs with moving interfaces/boundaries using unfitted finite element methods. We will describe the numerical discretisation of transient problems using unfitted finite elements that are robust with respect to the small cut cell problems. We will design these algorithms by extending our previous work to transient problems e.g. by defining space-time discrete extension operators.We will propose two different ways to design space-time unfitted methods. One approach is a pure space-time formulation in which our geometries are considered in 4D (for a 3D problem in space). This approach is suitable e.g. for problems in which the geometry is described via level sets. For complex geometrical representations in terms of oriented surface meshes we propose a geometrical discretisation framework (for 3D in space) that provides all the quadrature rules needed to integrate our numerical methods on unfitted meshes. The extension of this 3D algorithm to 4D is a challenge. We propose another space-time approach that solves the problem in the time-varying domain by using an extrusion of the 3D problem and a geometrical map. This way one does not require 4D geometrical algorithms. In time we consider discontinuous Galerkin spaces. The integration of inter-slab jump terms involves two functions on each side of the interface that are defined on different meshes (the background mesh and the mapped background mesh). In order to exactly compute these integrals we propose to use intersection algorithms.