We introduce the concept of unreliable failure detectors and study how they can be used to solve Consensus in asynchronous systems with crash failures. We characterise unreliable failure detectors in terms of two properties --- completeness and accuracy. We show that Consensus can be solved even with unreliable failure detectors that make an infinite number of mistakes, and determine which ones can be used to solve Consensus despite any number of crashes, and which ones require a majority of correct processes. We prove that Consensus and Atomic Broadcast are reducible to each other in asynchronous systems with crash failures; thus the above results also apply to Atomic Broadcast. A companion paper shows that one of the failure detectors introduced here is the weakest failure detector for solving Consensus [CHT92].