In this book the authors study the zero divisor graph and unit graph of a semigroup. The zero divisor graphs of semigroups Zn under multiplication is studied and characterized. The zero divisor graphs of the semigroups (Zp, ×), p a prime are only trees with p vertices. However the zero divisor graph of (Zn, ×), n a non prime is not a tree. We also define the special zero divisor graph of a semigroup and the S-zero divisor of a semigroup.