A ring R is called eversible if every left zero-divisor in R is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that R is eversible if and only if its upper triangular matrix ring T-n(R) is eversible, and if M-n(R) is eversible then R is eversible.