For more than four decades, scientists have been trying to find an answer to one of the most fundamental questions in paleoclimatology, the “faint young Sun problem.” For the early Earth, models of stellar evolution predict a solar energy input to the climate system that is about 25% lower than today. This would result in a completely frozen world over the first 2 billion years in the history of our planet if all other parameters controlling Earth's climate had been the same. Yet there is ample evidence for the presence of liquid surface water and even life in the Archean (3.8 to 2.5 billion years before present), so some effect (or effects) must have been compensating for the faint young Sun. A wide range of possible solutions have been suggested and explored during the last four decades, with most studies focusing on higher concentrations of atmospheric greenhouse gases like carbon dioxide, methane, or ammonia. All of these solutions present considerable difficulties, however, so the faint young Sun problem cannot be regarded as solved. Here I review research on the subject, including the latest suggestions for solutions of the faint young Sun problem and recent geochemical constraints on the composition of Earth's early atmosphere. Furthermore, I will outline the most promising directions for future research. In particular I would argue that both improved geochemical constraints on the state of the Archean climate system and numerical experiments with state-of-the-art climate models are required to finally assess what kept the oceans on the Archean Earth from freezing over completely.