  This work presents a reduced order modelling technique built on a high
fidelity embedded mesh finite element method. Such methods, and in particular
the CutFEM method, are attractive in the generation of projection-based reduced
order models thanks to their capabilities to seamlessly handle large
deformations of parametrized domains. The combination of embedded methods and
reduced order models allows us to obtain fast evaluation of parametrized
problems, avoiding remeshing as well as the reference domain formulation, often
used in the reduced order modelling for boundary fitted finite element
formulations. The resulting novel methodology is presented on linear elliptic
and Stokes problems, together with several test cases to assess its capability.
The role of a proper extension and transport of embedded solutions to a common
background is analyzed in detail.
