The behavior of the ultra-cold Bose gas is often intriguing and counter-intuitive. This applies particularly to the Bose gas under rotation, which shows superfluid effects such as quantized vortices and persistent currents. In my thesis, I investigate these phenomena by diagonalizing the Hamiltonian. Among the results we find that several quantities can be described by simple analytical expressions, and we discover a rich variation of vortex states. I will here present some of these results starting from a general physics approach.