This proposal brings together the fields of ultracold Fermi gases and of mesoscopic systems. Starting with a two-dimensional (2D) Fermi gas, we will imprint small-scale potential structures onto the atoms. Thus, a mesoscopic system embedded in a 2D reservoir is produced.Specifically, we will imprint optical dipole potentials varying on a micrometre scale onto a 2D gas of 6Li atoms. Due to the widely different energy scales, the entropy of the atoms in the mesoscopic structures will be massively reduced as compared to the reservoir atoms. The atoms in the mesoscopic structures will be characterised by an innovative detection scheme with single atom sensitivity. The combination of mesoscopic potentials, single atom detection and entropy reduction will put us in a unique position to access new regimes of many-body physics.First, we will investigate a mesoscopic realisation of the 2D Hubbard model. Beyond the study of the fermionic Mott insulating phase and its excitations, the possibility to study staggered Hubbard models and create domain structures is a very attractive prospect. Most importantly, the massive entropy reduction inherent to the mesoscopic approach will enable us to observe antiferromagnetic ordering, the major milestone central to further progress in the field.Going beyond periodic structures, we will focus on the direct creation of mesoscopic model systems. In a bottom-up approach, we will realise a plaquette consisting of 2x2 sites, the essential building block for models of d-wave superconductivity. The creation of 1D structures with local defects will open the possibility to study phenomena such as spin-charge separation, Friedel oscillations and the rectification of atomic transport. Finally, the physics of open quantum systems will become accessible when studying the interaction between mesoscopic system and reservoir. In conclusion, I believe that the proposed research programme will bring a new level of functionality to the field.