It is well-known that a ring with no nonzero nilpotent elements - a so-called reduced ring - is a subdirect product of domains. Moreover, as we have recently shown [2], a prime ring with only finitely many nilpotent elements is either a domain or is finite. In view of these results, it is natural to ask what can be said in general about rings with only finitefy many nilpotent elements. A crucial property of such rings is that they contain no infinite zero subrings, hence we are led to consider rings with this property also. Our principal result is that for any ring R with only finitely many nilpotent elementsis a direct sum of a reduced ring and a finite ring, where p(R) denotes the prime radical of R. One consequence is a finiteness theorem for periodic rings; another is the rather surprising result that every ring with infinitely many nilpotent elements has an infinite zero subring.