I am not an expert in this area and I am just translating material about French inheritance laws that I've found on French Wikipedia.
Under French law, your surviving spouse and your children, legitimate or illegitimate, from whatever marriage, cannot be disinherited — they have a status called « réservataire » when it comes to your estate. Each child has a right to inherit an equal part of your estate. Your surviving spouse can elect to take either a lifetime right to the whole estate, or to take outright ownership of a one-quarter share.
If you leave behind descendents when you die, those descendents take precedence over any other family members. If are survived by children, they divide up the inheritance in equal shares. However, if the one of your children has died before you die, any children of that child (your grandchildren) will inherit your deceased child's share.
If you have no descendents when you die, the inheritance will go to your other family members including ascendents (parents or grandparents) or collateral relatives (siblings, aunts and uncles, cousins).
If you are survived by your parents and/or siblings but no descendents, your mother and father each inherit one quarter of your estate and the remaining half goes to your siblings, each getting an equal share.If one of your parents is deceased, his or her one-quarter share is given to your siblings. If both your parents are deceased, the entirety of your estate is inherited by your siblings.
If you are not survived by your parents or any siblings, your estate is inherited by your surviving aunts, uncles, and cousins. One half of the estate goes to surviving relatives representing your maternal line and one half goes to those representing your paternal line. If all the relatives in one line or the other are deceased, the inheritance goes in its entirety to the surviving relatives of the other line.
As for your parents, siblings, aunts, uncles, and cousins, they can all be "disinherited" by drawing up a last will and testament that designates your surviving spouse as your sole heir. This is called a « donation entre époux » and has by law to be set up by a notaire in France.
Starting in August 2015, French inheritance laws will no longer be "unilateral". In other words, the inheritance rules and laws that apply in an individual's case will be those of the country where the death occurred. The law will apply to real property and all other assets.
There is also a provision explaining that your last place is residence is the criterion on which the decision about the laws regarding your estate is based, but that's not at all clear to me. What if my place of legal residence is in Saint-Aignan but I happen to die in an accident while I'm visiting my family in the U.S.? What inheritance laws would apply? Does having dual nationality play any role in all of this?
In France, when you inherit you inherit property and assets you also inherit the deceased's debts. Because of that fact, nobody can be required to accept an inheritance. If your deceased parent's debts amount to more than the value of the estate you would inherit, you can simply refuse the inheritance.
None of this deals with the issue of inheritance taxes, which are higher as your relation to the deceased is more distant. The surviving spouse and children inherit tax-free, I believe, but all other relatives pay significant inheritance taxes — up to 60% or more of the value of the property inherited. I'll have to look more into those issues. (See this post for more information.)
I think that all this is very interesting and that it might be helpful to you who now live in France or are thinking about moving here and buying property.