Abstarct: The power graph P(S) of a semigroup S is a graph whose vertex set is S and two vertices a, b ϵ S are adjacent if and only if a ≠ b and am = b or bm = a for some positive integer m. The power graph P(G) of a group G is defined similarly. In this thesis, we characterize the class of semigroups S for which P(S) is connected or complete. As a consequence, we prove that P(G) is connected for any finite group G and P(G) is complete if and only if G is a cyclic group of order 1 or pm, for some prime number p and for some m ϵ N. Then we compute the number of edges of P(G) for finite group G and we show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graphs. Also we discuss the planarity and vertex connectivity of the power graphs of finite cyclic, dihedral and dicyclic groups.