Abstarct: In the hat problem, each of n people is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color looking at the hat colors of the other people. The team wins if at least one person guesses his hat color correctly and no one guesses his hat color wrong, otherwise the team loses. The aim is to maximize the probability of winning. In this thesis, we cosider such problem on a graph, where vertices are people and a person can see all people, to which he is connected by an edge. We investigate the hat problem on the trees, paths, complete graphs and some cycles, and separately in each case obtain the hat number. Also we obtain the hat number on bicyclic graphs as our own work.