Moore graphs is a very rare class of graphs with an interesting properties. These graphs are extremal, that is, they have extremal properties: for fixed maximum degree and diameter, they have the largest possible number of vertices. There are only few Moore graphs, so we can count them in a few seconds. The main part of this diploma thesis is concerned with the analysis of properties of Moore graphs, and a special attention is given to Moore graph of type (3, 2) or Petersen graph. It is a very special graph because it appears very often as a counterexample when one tries to prove some statement. Moore graphs are regular graphs, which makes their construction not to difficult. The largest Moore graph which was successfully constructed is the Hoffman – Singleton graph, that is, the Moore graph of type (7, 2). This graph is 7-regular with 50 vertices and 175 edges. The biggest challenge in the study of Moore graphs is the construction of Moore graph (57, 2). Although this graph is arithmetically feasible, nobody has proved its existance. If such graph would exist, it would have 3250 vertices and 92625 edges. So the existance of Moore graph of type (57, 2) is still an open problem in graph theory. Since Moore graph have been discovered in 1960s, we can conclude that this part of graph theory is still ”young”, so there is enough time for progress in proving the existance of Moore graph (57, 2).