to happen: for an event to come about, usually one that lands on a particular person
Greek already had a general verb for something coming to be, γίνομαι, and the New Testament uses it hundreds of times. This word is narrower. It fastens an event to somebody: the trials that came on Paul, the healing the temple crowd saw happen to the beggar at the gate, the ordeal the scattered churches were living through. In nearly every use the Greek names the person on the receiving end.
That focus is why the word so often carries hard news. Jesus uses it for what is about to be done to him in Jerusalem, Paul for the plots he survived, Peter for a fire of testing his readers must not treat as a shock. It is not a gloomy word by nature, and it covers a good outcome just as easily, but the writers reach for it when something comes down on people rather than when the world simply turns.
Two other shapes turn up once each. In one, the past participle stands on its own for the events themselves, so the pair walking to Emmaus discuss "all these things that had happened" without saying whose they were. In the other, Luke writes the way Greek historians wrote, a plain "it turned out that," to report that the soldiers ended up carrying Paul up the barracks steps.