Your presentation is over, but now the audience has a chance to ask questions. This part is (even) less predictable then the presentation part, and therefore people get really scared about it (for no good reason).
Intended for: BSc, MSc, PhD
This is not an attack.
Don't get scared about questions. The people in the audience just want to engage a bit with you, are curious about something. They certainly don't know the answer right, otherwise they would not be asking. Stop seeing presentations as warfare, and be glad someone actually wants to engage.
Let the questioner finish.
Some presenters are so enthusiastic, they think they already understand the question after the first 5 words from the audience. This is really annoying, especially because you might be answering a different question then intended. Some questions take way too long, but a least give them the chance to finish their first few sentences. It actually gives you time to think about your answer as well.
Take your time to think.
Some questions a complicated, and you did not see them coming. That's totally fine. Just mention out loud that you need to think abou this for a second. It's totally fine if you are silent for 15 seconds just thinking. Wouldn't you do the same thing if you had this discussion with a colleague in a room? If you can't get to the answer at the spot, simply say so: "I would need to think about this for a bit long" or maybe "My first intuition says ...., but I will think about it a bit more."
Admit if you don't know.
Sometimes you just don't know. Simply say so, no problem. 90% of your audience has probably not even understood the question.
Don't give too long answers.
Again, you are super enthusiastic, but try to keep your answer within reasonable bounds. Some people continuously get distracted in their own answers, and they start to explain all other kinds of things. It mostly conveys that it is total chaos in your mind. Get to the point. If every answer to a question takes you 5 minutes, you are doing something wrong.
Propose to take the discussion offline.
Consider whether the discussion is still interesting to the broader audience. If you get into very specific details with another expert, propose to take the discussion offline (in person) after the talk. You then have time left for other more general questions. (Note that the sentence "let's take this question offline" is - I think - also used to escape questions the speaker cannot directly answer, avoiding item 4.)