This paper is based on material presented at the Research Institute for Mathematical Sciences (RIMS), Kyoto University, in October 2012 in a series of lectures. Thus, on one hand, it serves as the lecture note of this minicourse Combinatorial rigidity: graphs and matroids in the theory of rigid frameworks. On the other hand, this final, extended form is perhaps closer to a short monograph on combinatorial rigidity problems of two-dimensional frameworks. It contains the fundamental results of this area as well as a number of more recent results concerning extensions, variations and applications. Also added are several exercises and some new results.