In general, the rule is that the exemption goes to to parent with whom the child(ren) spends the majority of its (their) time. This can be modified by agreement and the filing of the proper forms. From the way you phrased the question I assume that the divorce decree was silent on this question. (If so, and you used a lawyer, you may end up with a malpractice claim against that lawyer). If you cannot tell who has a particular child a majority of the time (legal custody is mostly beside the point here) go directly to a tax professional - do not pass go; do not collect $200. You should probably consult a professional no matter what, at least until this all gets ironed out.

Let's assume that the children spend a majority of their time with you. Then you can claim the exemption (unless they supply more than half of their own support) and your filing status is Head of Household. You and your ex can allocate the children and the exemptions between yourselves, and can change it from year to year, so long as you file the proper paperwork.

A fairly common problem occurs when a parent not otherwise entitled to claim the children does in fact claim them and files first. This would lead to the rejection of the second's parent's return if e-filed. If you are in this situation, get help immediately.