bigamist:

a ’spouse’ = a wife to a man, or a husband to a woman

‘bigamy’ is the act of a man having two wives or a woman having two husbands at the same time, usually the first wife or husband not knowing about the second one, in countries or societies where having more than one wife or husband is illegal or crime

Note: A ‘bigamist’ is a person who is married to two people (spouses) at the same time (‘bi’ = two), usually the first one not knowing about the second or vice versa; (‘poly’ = many) a ‘polygynist‘ is a man with two or more wives at the same time, all the wives know about one another, normally live on the same premises and call themselves ‘sister wives’; and a ‘polyandrist‘ is a woman with two or more husbands at the same time, marrying all of them at the same time or one after the other, and all the husbands know about one another. A man or woman who has sexual relationship only with the person he/she is married to legally is called a ‘monogamist‘ (‘mono’ = single/one), but if a person has sexual relationship with people other than his/her spouse — the legally married person, those people are not called ‘wife’ or ‘husband’ of that person and the person is not considered a ‘bigamist’ or ‘polygamist’. The systems of  ’bigamy’, ‘polygamy’, ‘polygyny’ and ‘polyandry’ are culturally, socially and legally accepted by the society, but having sexual relationship with persons other than the legally married one is not accepted by any society — ancient or modern — and it is considered a crime. And there is this ‘polyamorist’, usually said ‘poly’ in short, is a person who believes and supports the idea of several people, men and women, having sexual or romantic relationships with one another with the approval/consent of each one involved.

However, though ‘bigamist’ is strictly refers to a person with two spouses, some people use ‘bigamist’ for a person having two or more spouses at the same time.