In this chapter, possible limitations of the hypothesis-testing approaches to the philosophy of science are outlined, in particular, that the theory-dependence of observations is a problem for falsificationism as well as for empiricism. How can we falsify a hypothesis by observations counter to predictions from the hypothesis if observations are fallible? The related Duhem-Quine problem is introduced: the design of experiments is invariably dependent on a number of theoretical assumptions, so a negative outcome may imply that the hypothesis is wrong but could just as well imply that any of the assumptions are wrong. The important role of confirmation of hypotheses, rather than falsification, is introduced, as well as the problem of how scientific progress can work in light of falsificationism - can science really progress by just proving ideas wrong? The problems are summed up, leading to Feyerabend’s general critique of the scientific method. The strong lasting influence of falsificationism on science today is acknowledged, and the alternative approach of Bayesianism is introduced.