In this thesis, the use of low-rank approximations in connection with problems in system identification is explored. Firstly, the motivation of using low-rank approximations in system identification is presented and the framework for low-rank optimization is derived. Secondly, three papers are presented where different problems in system identification are considered within the described low-rank framework. In paper A, a novel method involving the nuclear norm forestimating a Wiener model is introduced. As shown in the paper, this method performs better than existing methods in terms of finding an accurate model. In paper B and C, a group lasso framework is used to perform input selection in the model estimation which also is connected to the low rank framework. The model structures where these novel methods of input selection is used on are ARX models and state space models, respectively. As shown in the respective papers, these strategies of performing input selection perform better than existing methods in both terms of estimation and input selection.