Kate They are both correct - the one you use depends on context. "Do you want something?" is usually used to mean, "What are you doing here and what do you want?", or "Why are you disturbing me?" "Do you want anything?" is more often used when asking if someone wants a particular item.

Either or! "I'm going to the shop, do you want something?" and "I'm going to the shop, do you want anything?" are almost identical except that the first maybe implies a little bit more specificity.

to me it's the (almost) same, doesn't Russian do it too?

Think of your answer to each question: "Yes, I want something: a Big Mac" or "yes, I'd like anything, but more specifically I want a Big Mac."

"Do you want something?", might possibly come across as a bit rude, depending on the situation and the speaker's tone of voice. "Did you need something?", sounds more polite, firstly, because the past tense "did" implies that when you leave, your need will already have been taken care of, and secondly, because a person has more of a right to request "something needed", as opposed to "something merely wanted". "Do you want anything?" is also fairly polite, because it implies that the speaker is willing to provide whatever s/he can.