In this paper we shall first define the radical of a semigroup S and investigate some of its properties. Just as in the case of rings, one finds that the radical of a semigroup is a quasi-regular two-sided ideal which contains each quasi-regular right ideal of the semigroup and that the radical of a semigroup contains each nil right ideal of the semigroup. If a semigroup with zero satisfies the minimum condition on left ideals and right ideals then its radical is also the left radical of the semigroup. One also finds that if S. is the semigroup of all row-monomial n x n matrices over S then the radical of S,, is the semigroup of all row-monomial n x n matrices over the radical of S. If T is a two-sided ideal of S then the radical of T is the intersection of T and the radical of S. In the latter part of the paper we shall study semisimple semigroups. If s, t E S, let spt provided that, if m is contained in some 0-transitive operand of S, then ms = mt. Thus p is a two-sided congruence called the radical congruence of S. If the radical congruence of S is the identity relation then S is semisimple. Since M is a 0-transitive operand of S if and only if M is a 0-transitive operand of S/p, then it is easily seen that S/p is semisimple. If S is semisimple and T is a two-sided ideal of S, then T is semisimple. Finally, if S contains more than one element, then S is semisimple if and only if S is isomorphic to a subdirect sum of a set of semigroups each of which has a faithful 0-transitive operand.