What does the probate court deal with specifically?

Probate courts are specialized courts that primarily handle matters related to the administration of estates following a person's death. This includes validating wills, overseeing the distribution of assets in accordance with a deceased person's wishes, resolving any disputes regarding the interpretation of a will, and managing trusts. Essentially, the focus of probate court is on all legal issues that arise from wills and estates, which includes ensuring that debts are settled, assets are appropriately transferred to heirs, and the overall process abides by legal standards.

The other options, while important areas of law, do not fall within the jurisdiction of probate courts. Matters related to rental contracts are typically handled in civil courts, criminal litigation is addressed in criminal courts, and family law disputes, including matters of divorce and child custody, are managed in family courts. Thus, the probate court's specific role in handling the nuances of wills and estate matters clearly makes it the correct answer.