The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a^{m}=b or b^{m}=a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p^{m}. Particular attention is given to the multiplicative semigroup ℤ_{n} and its subgroup U_{n}, where G(U_{n}) is a major component of G(ℤ_{n}). It is proved that G(U_{n}) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U_{n}) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U_{n}) has no Hamiltonian cycle.