Graph-walking automata are finite automata walking on graphs given as an input; tree-walking automata and two-way finite automata are their well-known special cases. Graph-walking automata can be regarded both as a model of navigation in an unknown environment, and as a generic computing device, with the graph as the model of its memory. This paper presents the known results on these automata, ranging from their limitations in traversing graphs, studied already in the 1970s, to the recent work on the logical reversibility of their computations.