When can you start a sentence with the word "and" ?

I just took a "Teaching Writing to Elementary Students" cl and was told, to my great surprise, that at times it is acceptable to start a sentence with the word "and". I asked under which cirstances this was considered proper grammar, since it certainly was not when I was going to school, and was informed that there wasn't any time to go over that at that moment. After cl the teacher immediately left, so I am still wondering. Does anybody know? How about other conjunctions? Is there a grammatical situation, that anybody knows about, where it is grammatically correct to start a sentence with a conjunction?