Contemporary discussion of scepticism focuses on the possibility that most or all of our beliefs might be false. I argue that the hypothesis of massive falsity and the associated ‘problem of the external world’ are inessential to the scepticisms of Descartes and Hume. What drives Cartesian and Humean scepticism is the demand for certainty: any possibility of error, however local, must be ruled out before we can claim either justified belief or knowledge. Contemporary philosophers have ignored this form of scepticism because they doubt that the demand for certainty can be motivated. But Descartes provides a sound motivation for this demand in the Meditations.