If you beg someone to do something, you ask them very anxiously or eagerly to do it.

If someone who is poor is begging, they are asking people to give them food or money.

You say 'I beg to differ' when you are politely emphasizing that you disagree with someone.

If you say that something is going begging, you mean that it is available but no one is using it or accepting it.

If you say that something begs a particular question, you mean that it makes people want to ask that question; some people consider that this use is incorrect.

If you say that something begs a particular question, you mean that it assumes that the question has already been answered and so does not deal with it.