I would prefer not get a gift from someone that is dependent on me. I don't want to end up in a situation of perceived (by the person receiving the gift, giving the gift, or by an outsider) bribery. I realize that in many cases it is an innocent show of appreciation, but just saying (or writing) "thank you" will achieve the same goal without the risk of it being perceived the wrong way.

No way! It puts them in an uncomfortable position, even if they do not (and were not tempted to) accept it. They have the choice of (possibly) upsetting you, considering this a potential influencing attempt, even if not ultimately completed, in some places having to report it.

In our place, we have to report all gifts or gift attempts, and hand them to the institution if they exceed a certain (very small) value. In other words, it creates extra work and we don't even get to enjoy it.

Thank the professors that decided to take you on by doing a good job.

Oddly enough, the most appreciated gift from a student is probably just your thanks delivered personally. Even just a passing conversation in the hall way expressing your appreciation for their help will be remembered.

Students are often very appreciative of their professors, but often feel too embarrassed to actually say it. Of course, the prof will be embarrassed to hear it, but that is partly what makes it special.

We try hard, most of us, to help our students in every way we can. It is nice when that is noticed.

Giving a gift to someone in the department where you will be studying will be awkward and probably inappropriate. A thank you note or card would be more appropriate by far.

It makes more sense to give someone a gift when you are graduating and leaving the program. That expresses your appreciation more concretely while not having the appearance of trying to curry favor with a superior or repay some kind of inappropriate favor.

I also gave gifts to people who wrote me letters of recommendation, but they were very small gifts (like a chocolate bar with a thank you note).

Write thank-you letters to those professors who interviewed you. Expressing your appreciation for their time is more than appropriate, but offering a gift at this early stage is not. Reserve gifts for mentors with whom you have established relationships and, even then, only for special occasions (perhaps holidays or the end of the academic year, for example). Otherwise, you’re creating a situation that can too easily be misconstrued as a conflict of interest.

As a PI (in France though) I would say that no, this is definitely not good idea for all the reasons mentioned in the other answers.

On the other hand, bringing something with a low economical value for the team - and not only for the professor, this is important - would certainly be appreciated. Like a box of chocolates or, as suggested in the comments some culinary specialty from your own country/region if it's different from the lab.

Again, the important point is that it is shared by the whole group.

We have a somehow similar habit in our lab (approx 50 people) : when someone goes on a conference or on holidays abroad he/she usually bring back a specialty to share for everyone. No one is forced to do so, but this is usually highly appreciated and almost everybody does it.

A close friend who is a professor often has Ph.D. students from Korea and China, who give small gifts--food items from their home countries, for example. (We both work in the U.S.; I am also a professor, although in a different field.) I don't think I've heard of American or European students doing this sort of thing. My intuitions agree with other answers that warn against physical gifts. However, based on my friend's experience, my guess is that little gifts for professors are common, and considered appropriate, in Korea and China. If so, then the correct answer may depend on what's appropriate in the country in which you'll be studying--Germany, in your case. Perhaps other people can specifically respond for that case. (My sense is that the academic system in Germany is different from that in many other European countries, so it's possible that the culture is different, too. It could be that there is a different answer that's appropriate for Germany rather than, say the U.K., France, or the Netherlands.)

Just thank now, no gift, as most of the other answers say. I'll add "thank later" too. The best gifts I've gotten from students come years after I taught them, when they remember something I helped them with and they write to say so.

I've never seen a case of this being done, and I think this is generally a bad idea. The professor hasn't yet really done anything for you to be thankful for and there's a real risk of it being perceived as apple-polishing (bribery would be too strong a word). As a general rule, starting a relationship with a gift makes things awkward without any substantial benefit to anybody.

If you are inclined to give something physical to your new boss, I would suggest you follow @lalala's suggestion and `a few snacks from your home country to the group' or bring some sort of souvenir that connects with your background (e.g., if you were involved in a student association and they had great T-shirts, you could bring one). A reasonable litmus test would be: Would you be comfortable gifting the same thing to your peers?

Also, feel free to give them a gift after you finish your degree. The dynamic is very different then: They stop having any real power over you and they have had a major impact on your life.

As a former professor (USA) (now a full time research scientist) let me add to the chorus: NO GIFTS, EVER.

Gifts to a superior are a propitiation; I would not only refuse gifts from a student but report the attempt to my department chair, so I wouldn't be compromised by not reporting an offer of a bribe in the future. I would probably have to rescind my offer to advise you, and let you find somebody else.

Your urge to give a gift is called propitiation and it should be quashed; it IS a bribe to gain favor, not thanks. If I selected you, it was not to do you a favor, but because I thought with my management you could make a contribution to our field. If I didn't think that, I would spend my time on someone else, or on my own research.

Saying thanks is enough. Don't offer to buy me lunch, don't offer me tickets to something, don't try to use money or anything else of value to establish a personal relationship with me. It is possible we WILL become friends by the time you are ready to defend. I became (and remained) friends with my advisor. If that is going to happen it will be a result of shared cultural interests without any gifts being exchanged.

Yes!

I recommend gifts such as your attention, ideas of your own and questions...lots of questions. No physical gifts though, those aren't necessary.