  In this chapter we examine reduced order techniques for geometrical parametrized heat exchange systems, Poisson, and flows based on Stokes, steady and unsteady incompressible Navier-Stokes and Cahn-Hilliard problems. The full order finite element methods, employed in an embedded and/or immersed geometry framework, are the Shifted Boundary (SBM) and the Cut elements (CutFEM) methodologies, with applications mainly focused in fluids. We start by introducing the Nitsche's method, for both SBM/CutFEM and parametrized physical problems as well as the high fidelity approximation. We continue with the full order parameterized Nitsche shifted boundary variational weak formulation, and the reduced order modeling ideas based on a Proper Orthogonal Decomposition Galerkin method and geometrical parametrization, quoting the main differences and advantages with respect to a reference domain approach used for classical finite element methods, while stability issues may overcome employing supremizer enrichment methodologies. Numerical experiments verify the efficiency of the introduced ``hello world'' problems considering reduced order results in several cases for one, two, three and four dimensional geometrical kind of parametrization. We investigate execution times, and we illustrate transport methods and improvements. A list of important references related to unfitted methods and reduced order modeling are [11, 8, 9, 10, 7, 6, 12]. 
Reduced Basis, Embedded Methods and Parametrized Levelset Geometry