When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer a significant advantage in dealing with complex geometries, eliminating the need for generating unstructured body-fitted meshes. However, current unfitted finite elements on nonlinear geometries are restricted to implicit (possibly high-order) level set geometries. In this work, we introduce a novel automatic computational pipeline to approximate solutions of partial differential equations on domains defined by explicit linear and nonlinear boundary representations. For the geometrical discretization, we propose a novel algorithm for generating quadratures for the bulk and surface integration on linear and nonlinear polytopes required to compute all the terms in unfitted finite element methods. The algorithms for generating these polytopes rely on linear geometrical kernels, Bernstein-B\'ezier representations and robust polynomial multivariable root-finding techniques .Furthermore, we propose a space-time approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefit of the proposed formulation is the ability to handle large displacements of the domain boundaries. To deal with the mesh movement, we pull back the equations to a reference configuration that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we consider an exact transfer operator to compute jump terms in the time discontinuous Garlerkin integration. The linear framework has been applied with success on all analysis-suitable oriented boundary meshes (almost 5,000) in the Thingi10K database. We successfully simulated partial differential equations with unfitted finite elements on nonlinear domains described by computer-aided design models. We also simulated fluid problems around rotating (2D and 3D) explicit geometries. In both approaches, we achieved optimal convergence of the numerical experiments.