We study Hubbard-type model in this thesis. In the first part of the thesis, we study exactly solvable Hubbard type models and construct two classes of exactly solvable models. In the first class, Jordan-Wigner transformation and spin rotation are applied to map different interacting fermion and spin models in one dimension. Bond-charge interaction and three body interaction between fermions are generated naturally with Jordan-Wigner transformation in these models. The models can be exactly solved at certain symmetric points. The second class is called Majorana Falicov-Kimball model which can be exactly solved in arbitrary dimensions. The construction of the exact solution is parallel to the exactly solvable Kitaev honeycomb model for S = 1=2 quantum spins and can be viewed as a generalization of Kitaev’s construction to S = 1=2 interacting lattice fermions. A BCS-Hubbard model is studied in detail in the thesis. We show that the model becomes exactly solvable for arbitrary U when the BCS pairing amplitude Δ equals the hopping amplitude t. The nature of the solution is described in detail. In the second part of the thesis, we study doped HgTe/CdTe quantum well with Hubbard-type interaction under perpendicular magnetic field using a lattice Bernevig-Hughes-Zhang (BHZ) model with a bulk inversion asymmetry (BIA) term. Within a simple mean-field theory we show that the BIA term is strongly enhanced by interaction around the region when the band inversion of the topological insulator is destroyed by a magnetic field. The enhanced BIA term creates edge-like electronic states which can explain the experimentally discovered edge conductance in doped HgTe/CdTe quantum well at similar magnetic field regime by Shen et al..