Most courts give great deference to the parents of children in child custody matters, and grandparents are awarded custody of their grandchildren only under limited circumstances.

For example, if both parents are unfit to have custody of their children, or if both parents are deceased, or if one parent is deceased and the other is in prison, grandparents may petition the court for custody of their grandchildren.

And grandparent visitation rights vary from state to state, and are often conditioned on certain circumstances, such as when the child’s parents are divorced, separated, or deceased.

In Connecticut, like in many states, the primary consideration in child custody matters is the best interest of the child. Parents are generally presumed to be fit custodians, and courts typically defer to the parents' rights to custody. However, under certain circumstances, grandparents may be awarded custody if both parents are deemed unfit, if both parents have passed away, or if one parent is deceased and the other is incarcerated. Grandparents can petition the court for custody in such situations. Regarding visitation rights, Connecticut law does allow grandparents to petition for visitation rights under specific conditions, such as when the parents are divorced, legally separated, or deceased. The court will consider whether such visitation is in the best interest of the child, taking into account the child's needs and the circumstances of the case. It's important to note that these matters are complex and outcomes can vary widely based on the individual facts of each case.