I am studying at Computer Science.

What is the best way to ask a student what he/she studies? Both are acceptable! I also say "what are you studying?" There are many ways to ask the question.

"what do you study?" is okey, it's a general question, also "What are you studying?" is correct and from my point of view it is kinder than the first one, polite.

At the University level, in the US, "what is your major" is the more commonly used phrase (i.e.; I'm majoring in Anthropology and minoring in Linguistics). "What do you study" tends to bring on a list of interests or current classes (i.e.; "I'm taking polisci, English 110, Math 111, Comm 122, and Underwater Basketweaving). Based on Paul's comment, it seems like it depends on whether you are in the UK or the US. Hope this helps!

We don't have majors and minors in the UK so I'd go with the first question here. We understand what 'what is your major?' means but like I said, due to the different education system nearly everyone studies one subject from day one of their course.