"Related" list of PS4 controller troubles on a question about PS4 controllers

Somehow, many people have trouble with connecting controllers to their PS4, and new questions asking how to solve the issue are still being asked.
It would be helpful to have a single question and answer listing the most common causes of connectivity problems and the steps to solve them, similarly to I can't play Pokemon GO! What's wrong?.

(If we already have such a question, please link it here to make it easy to find when casting close votes)

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