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I discuss a lattice field theory investigation of the phase structure of the 2+1-dimensional Gross-Neveu model in the large-$N_f$ limit, where $N_f $denotes the number of fermion flavors. Numerical results are presented, which indicate the existence of an inhomogeneous phase at small temperature and large chemical potential similar as in the 1+1-dimensional Gross-Neveu model in the large-$N_f$ limit. I also discuss a lattice field theory simulation of the 1+1-dimensional Gross-Neveu model at finite number of fermion flavors. Also here evidence for the existence of an inhomogeneous phase is found.