Wait-free implementations of shared objects tolerate the failure of processes, but not the failure of base objects from which they are implemented. We consider the problem of implementing shared objects that tolerate the failure of both processes and base objects. We identify two classes of object failures: responsive and non-responsive. With responsive failures, a faulty object responds to every operation, but its responses may be incorrect. With non-responsive failures, a faulty object may also "hang" without responding. In each class, we define crash, omission, and arbitrary modes of failure. We show that all responsive failure modes can be tolerated. More precisely, for all responsive failure modes F, object types T, and t, we show how to implement a shared object of type T which is t-tolerant for F. Such an object remains correct and wait-free even if up to t base objects fail according to F. In contrast to responsive failures, we show that even the most benign non-responsive failure mode cannot be tolerated. We also show that randomization can be used to circumvent this impossibility result. Graceful degradation is a desirable property of fault-tolerant implementations: the implemented object never fails more severely than the base objects it is derived from, even if all the base objects fail. For several failure modes, we show whether this property can be achieved, and, if so, how.