This licentiate thesis concerns topics in non-interacting and interacting quantum physics in one dimension. We present the notions of Wannier functions and tight-binding models. Quantum walks are discussed, quantum mechanical analogues to random walks. We demonstrate the ideas of Bloch oscillation and super-Bloch oscillation - revivals of quantum states for particles in a periodic lattice subject to a constant force. Next, the Rabi model of light-matter interaction is derived. The concept of quantum phase transitions is presented for the Dicke model of superradiance. The idea of adiabatic elimination is used to highlight the connectedness of the Dicke model. Finally, we present a one-dimensional interacting system of resonators and artificial atoms that could be built as a superconducting circuit. Using adiabatic elimination as well as matrix product states, we find the phase diagram of this model.