let g=(v,e) be a graph with vertex set v and edge set e.for two vertices u,v of g ,the closed interval i[u,v] ,consists of u,v and all vertices lying in some u-v geodesic in g.if s is a set of vertices of g then i[s]is the union of all sets i[u,v]for u,v ? s. if i[s]=v(g) , then s is a geodetic set for g.the geodetic number g(g) is the minimum cardinality of geodetic set.the maximum cardinality of a minimal geodetic set is the upper geodetic number g^+ (g). a set s?v(g) is a connected gcedeodetic set of g if i[s]=v(g) and the subgraph in g induced by s is connected.the minimm cardinality of a connected geodetic set of g is the connected geodetic number g_c (g)of g and a connected geodetic set of g whose cardinality equals g_c (g)is a minimum connected geodetic set of g. in this thesis, we study geodetic number and upper geodetic number of graphs.