*Paul Feyerabend (1924-1994) studied astronomy and physics but ended up a philosopher. He is not among the most well known philosopher of science, but I believe he is among those that come right after.

In "Against Method", he argues there are no universal methodological rules that consistently lead to scientific progress. In his words "the only principle that does not inhibit progress is: anything goes."

I should add he was highly critical of the mainstream views of science, energetically so.

Against the naive view of the scientific method

Here is a view of science Feyerabend deeply disagrees with (my words):

Scientists observe phenomena, form hypotheses about how things work, design and conduct experiments to test their ideas, analyze the results, and draw conclusions. If a hypotheses goes through many such cycles without being opposed by experiments it is considered valid for the time being. If a hypotheses is contradicted by an experiment, it is discarded. 
This cyclical process continuously improves our hypotheses and bring us closer to truth.

In Feyerabend's eyes this is neither a good description of what scientists are actually doing nor a good idea.

A large part of the book is dedicated to a case study on Galileo's scientific work and its reception by the intellectual and religious institutions of the time. Through this case study Feyerabend argues that sufficiently new theories would always be rejected if the rules above were applied. In the case of Galileo it was necessary to invent many things before any of his new proposals could belong to a coherent whole: new instruments of measure, a new theory of optics, a new theory of cosmology, and a new theory of physics. 
The new model of physics needs the new cosmology, otherwise the movements of "celestial bodies" disproves it.
Likewise the new cosmology needs the new model of physics, otherwise it is "physically impossible".
This plays into another important idea of this book: observations are never objective and do not exist outside of theories.

Against objective observation

Imagine there is a single consensual theory on a topic, we can call it theory A. 
Then somebody comes up with theory B, which is not compatible with A. It uses different primitive concepts, it cannot be used to predict what A predicts, and its own predictions don't really make sense within theory A.
According to Feyerabend, proponents of theory A will use these differences to discredit theory B.  Not realizing the framework they are used to carry its own assumptions, they will turn a simple proof of incompatibility between theories into a proof theory B is false. 
But nothing here says A is better, it simply came first. Again Feyerabend gives the example of Galileo's theories and how they made no sense in the context of what people at the time thought they knew.

All observations and judgements need a theoretical framework.
In other words, all observations are theory laden (Feyerabend goes a bit further than this but the distinction doesn't really matter).

A solution might be to ask that theory B always explain strictly more things than theory A. Any topic on which theory A can make predictions should be covered in theory B. This way we can always compare the two.
Feyerabend raises the objection that if we applied this rule consistently we would still be incapable of moving away from ambiguous and fuzzy theories of all of reality from antiquity, because we would need a fully coherent theory of reality to replace them. 
Indeed I think it is reasonable to propose narrower yet more complete theories to replace larger yet imprecise theories.

Against (all?) methods

Feyerabend goes much further than this opposition to the naive view of the scientific method described above. 
He is against all absolute rules and methods in science. In his words:

given any rule, however 'fundamental' or 'rational', there are always circumstances when it is advisable not only to ignore the rule, but to adopt its opposite.

I am not sure exactly how he claims to have proven this except by example and vibe, but it relates to the most important idea of the book: we cannot accept any strict rule over science, otherwise we won't be able to change perspective when a sufficiently new and better one comes up. For Lesswrong style rationalists, we might consider this a particular case of the principe of equal and opposite advice.

This is how Feyerabend justifies that in research "anything goes".

This does not mean there can be no useful rule or method or that they have no place in science.

[I believe] that both absolute rules and context dependent rules have their limits [I do not believe that] all rules and standards are worthless and should be given up

Feyerabend does not give a proper definition of what he means by "method".
Per my understanding, being "against method" means he objects to any attempt at saying "here is how to do science, everything else is to be rejected". He also objects if one only says instead "here is how to do aspect X of Y in science, everything else is to be rejected". For example, he wouldn't like "in machine learning, one must always accompany new algorithms with experiments featuring both a measure of precision and recall".

Against the reader

This book has flaws. 
It is not really unreadable. There are no grammar issues, the sentences are well formed, there is some structure. But according to Wikipedia Feyerabend himself called it a collage and, well, it shows. There are weird repetitions and especially toward the end it becomes difficult to see what the book is building towards. There are a few core ideas (mostly the ones listed above) and then there are a lot of more or less relevant digressions, many of which are very interesting.

Another issue is that Feyerabend enjoys provocation. This means he uses some hyperbole and surprising phrasing that are only there for shock value, even if they force the reader to do a double take. There is also some inflammatory speech spread out throughout the book, no doubt in the hope of annoying some specific other philosophers of science of his time that he disliked. At first I found it fun and endearing but over time it became grating.

Here are a few examples of some of the worst the book has to offer in terms of provocative tone

mystics, using only their minds, traveled across the celestial spheres to God himself, whom they viewed in all his splendour, receiving strength for continuing their lives and enlightenment for themselves and their fellow men.

(I am convinced the passage above was only included to make fun of those who accused Feyerabend of being anti-scientific)

science can stand on its own feet and does not need any help from rationalists, secular humanists, Marxists and similar religious movements  

Both cases rest on antediluvian concepts (and inadequate distinctions). Small wonder our 'rationalists' are fascinated by them.

Also the book keeps insulting "rationalists". But even if you identify with the label in 2025 you do not need to feel insulted. I am not sure whom he means exactly, but if the accusations he levies against them are even remotely fair (which is dubious) then their ideas had little in common with Lesswrong style rationalism. I expect that Karl Popper and Imre Lakatos would count as rationalists for the purpose of Feyerabend's criticism. But I am not sure there is a clear philosophical movement that matches with his criticism, though he does not seem to like secular humanism. In particular, I doubt whether Bertrand Russel was among those Feyerabend had in mind.

Does the scientific method exists?

I do think Feyerabend is right about many things. He is excessive in many ways and lacks rigor, but his main point has value and I agree with some of his ideas.

Like him I am not convinced that there is a clear universal method to tell scientist and the scientific community when it is time to change their mind, to discard a theory and adopt a new one. We do not have a global recipe to follow, a protocol that guarantees science will always progress if we follow its clear instructions. I am not certain there cannot be such a protocol in theory, but I don't think we are currently following one.

On this topic, I strongly recommend the book Science Fictions by Stuart Richie which discusses different failures of the academic systems with a focus on the ones that came to light as a part of the replication crisis. It is a better book, clearer and better thought out, but perhaps it does not lead to ideas as interesting as Against Method; it does not discuss the validity or definition of the "scientific method" itself.

Of course science not having a clear recipe for success does not mean it is not producing many great inventions and discoveries. There would certainly be much more to say on this topic but I do not want this review to focus on my own views.

Is this book any good? Should you read it?

You certainly can, it is full of interesting ideas, remarks, and arguments. If you care about the topic you might find it an interesting read. There is an amazing book to write out of this one with more structure, more rigor, and less redundancy. As it stands there were parts that felt like a chore but also many other parts I truly enjoyed. At least the book was intellectually stimulating.

If you do read it you should start by reading chapter 20, which is a bonus chapter of later editions in which Feyerabend goes over the life path that led him to his views. It is much more pleasant and engaging than the rest of the book and probably helps understand everything else better. You can read it online here (chapter 20 starts at page 252).*