We introduce the set S(R) of “strong zero-divisors” in a ring R and prove that: if S(R) is finite, then R is either finite or a prime ring. When certain sets of ideals have ACC or DCC, we show that either S(R) = R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. This is a finite union when R is a Noetherian ring. For a ring R with |S(R)| = p, a prime number, we characterize R for S(R) to be an ideal. Moreover R is completely characterized when R is a ring with identity and S(R) is an ideal with p2 elements. We then consider rings R for which S(R) = Z(R), the set of zero-divisors, and determine strong zero-divisors of matrix rings over commutative rings with identity.