The Popper-Kuhn debate generated several important successor positions.
Imre Lakatos (1922–1974) tried to reconcile Popperian falsificationism with Kuhnian realism about scientific practice. His concept of research programmes is more realistic than Popper's naive falsificationism. A research programme has a hard core of theoretical commitments that scientists treat as definitively established and protect from falsification, and a protective belt of auxiliary hypotheses that can be modified or discarded when facing recalcitrant evidence. When data conflicts with predictions, scientists modify the protective belt, not the hard core.
For example: when early telescopic observations seemed to conflict with Newtonian mechanics (the planet Uranus wasn't where it should be), astronomers didn't abandon Newton. They hypothesized an unknown planet (Neptune) whose gravitational effects were disturbing Uranus. The hard core (Newton's laws) was protected by adding a new auxiliary hypothesis (there's another planet). This turns out to be perfectly rational, and it worked, because Neptune was found.
Lakatos distinguishes progressive research programmes (the protected hard core keeps generating new, successful predictions) from degenerating ones (the protective belt keeps getting patched to accommodate failures without generating new predictions). Science is rational, on his view, but it operates at the level of programmes over time, not individual experiments.
Paul Feyerabend (1924–1994), Popper's former student and most entertaining critic, took the opposite direction. His Against Method (1975) argues that there is no single scientific method. Looking at the history of science, every proposed methodological rule, including falsificationism, has been productively violated at some point by scientists who turned out to be right. Galileo held his heliocentric theory in the face of apparently refuting observational data (the telescopic observations had serious accuracy problems). Darwin published the Origin of Species without a theory of inheritance (the mechanism for heredity was unknown). If they had followed Popper's rules, they might have abandoned correct theories too soon.
Feyerabend's conclusion: the only methodological principle that doesn't impede scientific progress is "anything goes", scientists should be free to use whatever methods seem productive, including proliferating incompatible theories, using propaganda and rhetoric, and violating established methodological norms. He called his view epistemological anarchism.
This sounds nihilistic but Feyerabend's actual target was more specific: the authority of science as a culturally privileged institution. He argued that Western science had no more epistemic authority than other knowledge traditions, that treating it as the uniquely legitimate form of inquiry was a form of cultural imperialism. This argument, whatever its merits, has had enormous influence beyond philosophy, in science studies, cultural theory, and philosophy of indigenous knowledge.
The contemporary consensus, if there is one, is somewhere between Popper and Kuhn: science does work through conjecture and refutation, but the process is messier, more social, and more historically contingent than Popper imagined. Falsifiability remains a useful heuristic for demarcation, but no single algorithmic method fully describes how science actually progresses. And the question of whether scientific knowledge is objective, or whether it is partly constructed by the communities that produce it, remains live.