Human–animal marriage
There are numerous cases of people seeking to marry animals they loved, from the Roman Emperor Caligula who married his horse and appointed it to the Senate, through to the 2005 case of a woman who married a dolphin, and a 2006 case of a Sudanese man who was ruled to be obligated to pay dowry and marry a goat following sexual activity with it. In addition there is significant folklore and myth on the subject. Possible some less well known religions, or older traditions, may indeed permit such marriages.
No modern country recognizes marriage between humans and non-humans, but throughout history people have sought ways to commit to an animal they loved in this way, often involving ceremonies which, whilst not legally bidning, carry great personal significance.
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Law
No current, national, jurisdiction allows marriage between humans and non-humans. They are not legally recognized and purported marriages have no validity. Additionally, in many customs, a marriage must be consummated, which in such a partnership would in many territories constitute an illegal act.