A personal union is the combination by which two or more different states have the same monarch while their boundaries, their laws and their interests remain distinct. It differs from a federation in that each constituent state has an independent government, whereas a federal state is united by a central government. The ruler in a personal union need not be a hereditary monarch.
Personal unions can arise for very different reasons, ranging from coincidence (a woman who is already married to a king becomes queen regnant, and their child inherits the crown of both countries) to virtual annexation (where a personal union sometimes was seen as a means of preventing uprisings). They can also be codified (i.e., the constitutions of the states clearly express that they shall share the same person as head of state) or non-codified, in which case they can easily be broken (e.g., by the death of the monarch when the two states have different succession laws).
^In the Holy Roman Empire, many prince-bishops had themselves elected to separate prince-bishoprics, that they ruled in a personal union. For example, Joseph Clemens von Bayern (1671-1723) was Prince-Bishop of Freising (1685-1694), Prince-Bishop of Regensburg (1685-1694), Prince-Elector of Cologne (1688-1723), Prince-Bishop of Liège (1694-1723) and Prince-Bishop of Hildesheim (1702-1723).