This paper introduces the notion of stability for a long-lived consensus system. This notion reflects how sensitive to changes the decisions of the system are, from one invocation of the consensus algorithm to the next, with respect to input changes. Stable long-lived consensus systems are proposed, and tight lower bounds on the achievable stability are proved, for several different scenarios. The scenarios include systems that keep memory from one invocation of consensus to the next versus memoryless systems; systems that take their decisions based on the number of different inputs but not on the source identities of those inputs versus non-symmetric systems. These results intend to study essential aspects of stability, and hence are independent of specific models of distributed computing. Applications to particular self-stabilizing asynchronous systems and synchronous systems are described.