What distinguishes science from all other human endeavours is that the accounts of the world that our best, mature sciences deliver are strongly supported by evidence and this evidence gives us the strongest reason to believe them.' That anyway is what is said at the beginning of the advertisement for a conference on induction at a celebrated British seat of learning in 2007. It shows how much critical rationalists still have to do to make known the message of Logik der Forschung concerning what empirical evidence is able to do and what it does. This paper will focus not on these tasks of popularization faced by critical ration- alists, but on some logical problems internal to critical rationalism. Although we are rightly proud of having the only house in the neighbourhood that is logically watertight, we should be aware that not everything inside is in impeccable order. There are criticisms that have not yet been adequately met, and questions that have not yet been adequately answered. Each of the six di-culties to be discussed arises from Popper's exemplary solutions to the problems of demarcation and induction. They concern the management of contradictions; approximation to truth; the corroboration of already falsifled hypotheses; decision making under un- certainty; the role of evidence in the law; and the representation of logical content. In none of these areas does critical rationalism yet ofier, to my mind, an account comparable in clarity to its solutions to the problems of demarcation and induction. This is a personal selection, and it is not suggested that there are not other hard questions ahead. In only one or two cases shall I ofier anything like a solution.