Polygamy (more specifically, polygyny) is a Muslim-exclusive (without Holy Fury DLC) marriage mechanic which allows Muslim men to marry up to 4 wives.
Concubinage is a similar, but distinct system used by Pagans, Zoroastrians, followers of the Eastern religions and all tribal and nomadic rulers (no matter their religion).
It should be noted that Muslim women are forbidden to marry more than one husband, even for female rulers. Similarly, without 
Holy Fury, women can never take male concubines.
Mechanics
Polygamy allows Muslims to bear a lot more legitimate children than Christians do. This is closely tied to 
decadence as it encourages plenty of sons who may become decadent, and therefore, they may need titles to be granted to.
There is a first wife who is the favorite among the wives. Stats boost to state attributes are received from the first wife only. However, secondary wives are still eligible to become the spymaster in the council.
Effects of polygamy
A Muslim ruler who does not marry enough wives will get a prestige penalty modifier of -1 point per missing wife. The expected minimum increases by one wife per title rank:
- A count-tier ruler (e.g. sheikh, marzoban, bey) should marry at least 2 wives
- A duke-tier ruler (e.g. emir, satrap, beylerbey) is expected to marry 3 wives, and
- a king-tier (sultan, shah) or emperor-tier ruler (badshah, shahanshah, padishah) should marry 4 wives.
Characters under polygamy tradition are encouraged to take as many wives as possible, and will even get a +0.4 prestige bonus per extra wife according to rank.
Choosing the right wives
With open-agnatic succession, the most powerful son or grandson, i.e. the one with the most land, will inherit. This allows for controlled breeding. You can easily get a genius as your successor when you marry only women with the 
trait. Suitable brides can be found using the character finder.
Muslim succession is agnatic and thus wives' claims are not pushable. However, sons can still inherit claims from their mothers.
Since Muslims do not gain or loss prestige based on marriage rank (if both bride and groom are Muslims), you are free to marry down (e.g. to breed geniuses). At the same time, you are also free to marry up (e.g. to gain powerful allies or claims), since the ruler with which you are negotiating is not concerned about their own prestige loss. However, rulers may still reject marriage offers as they might "want a better alliance/non-aggression pact".
Alliances
Alliances can be formed with the relatives of all wives (and the relatives of all your sons' wives), not just the first wife. Muslim rulers thus have it easier to create alliances, but may also entangle them in more wars as a consequence.
A Muslim ruler will also have it easier to placate vassals with their many marriage possibilities. Non-aggression pacts upon marriage make marrying within the realm more attractive: a vassal cannot join or form a faction against his liege when he has a marriage pact with him, nor can the liege revoke an allied vassal's title. This can be very good for realm stability if one maintains a larger number of vassals, such as the Umayyad and Abbasid dynasties in the early bookmarks.
Rivalry between wives
The first wife is a very prestigious title and the secondary wives will try to attain this status. They will sometimes ask the husband for the status of first wife, and if they are in a scheming mood, they might kill the present first wife.
Wives also like to see their offspring on the throne and might attempt to kill the current heir in favor of their own. They might even be bold enough to murder all sons but theirs. Wives who are 
, 
, or 
are more likely to attempt murder.
Age of Wives
There are some advantages in marrying at least 1 or 2 wives who are near the age limit of childbearing. (The maximum age of a bride for marriages arranged via the "rings" button is 40 years old; use the character finder to find older brides.) Apart from the prestige bonus, or prevention of prestige penalties, childless wives past childbearing age do not attempt murder on stepsons. With sons coming from one or two mothers, the heir is more likely to have full brothers and only suffer from -20 opinion penalty, rather than half-brothers who suffers from -40 opinion penalty as potential competitors.
Concubinage
Pagans (except Hellenic), Zoroastrians, followers of the 
Eastern religions, and all tribal and 
nomadic rulers (regardless of religion), rather than marrying multiple wives, may take up to three concubines. Muslim tribal and nomadic rulers cannot ask for concubines from iqta realms; they have to invite the woman to their court first.
Unlike secondary wives,
- a character can take a concubine without being married (This is useful if a wife of suitable status is unavailable, which would force a ruler to take the prestige penalties of marrying under your rank),
- a character can set aside a concubine, facing only a -20 opinion penalty from the woman set aside,
- at birth, children of concubines get the
trait that gives -1 Diplomacy, similar to bastards, but they take their place in the line of succession just like those of wives.
Rulers get a bonus to monthly Prestige for having "nubile concubines" i.e. still of childbearing age. Taking a concubine of high status (such as the daughter of a king or queen) gives +25 prestige per rank. Concubinage doesn't give rise to alliances.
A decent strategy is to marry politically and take concubines who have good genetic traits to produce heirs, to get good tutors for your children or to get a good spymaster (only for pagans). This is especially true for Zoroastrians and other divine marriage religions: marrying a close relative provides opinion bonuses with vassals, but also runs the risk of inbreeding. Having children with a concubine first, or marrying someone past childbearing age, allows for outbred heirs. Furthermore, your children will have the option of marrying their half-siblings, which is safer than marrying full siblings.
Lovers may be taken as concubines at the start of affairs to prevent any problems with siring bastard children or relation penalties to legitimate wives.
As of patch 2.8, it is possible to offer women as concubines to other characters.
Abduction
Women who are prisoners can be forced to become concubines. The abducted woman will be very angry (-50 "abductor"), along with her family (-25 "abducted close relative") and, if she was married, her former husband (-25 "wife snatcher").
If set aside, such women will stay in the abductor's court, allowing them to be married off to courtiers, although they will return home if their new spouse dies. Marrying these captured concubines to your own children can enable alliances and non-aggression pacts, even between rulers of different faith.
Relatives of an abducted woman can use the intrigue focus to attempt to free her, using the special interaction "Free from captivity". Owners of the 
Jade Dragon DLC may also use the "Free Captives" CB to free any unlanded hostages and forced concubines.
With Holy Fury
With Holy Fury, when a pagan religion is being reformed, the reformer can choose doctrines allowing followers of the reformed faith to practise marriage with multiple partners (polygyny or polyandry).
- Polygamy: Polygamy is allowed, but the possibility for a woman to take consorts is disallowed even if other traits should unlock it.
- Enatic Clans: Allows a woman to take up to 3 consorts, but places restrictions on the role of men.
- Equality: A woman can take up to 3 consorts, but enforces equality between the two sexes.
Note that female fertility ends at age 46; this, along with the fact that each pregnancy usually only produces one child, means that female rulers have less heirs even with multiple male consorts. Also, remember that female rulers can die during childbirth or be incapacitated by a Troubled Pregnancy, potentially causing a regency and complications in succession. In addition, the Final Months of a pregnancy will always trigger a regency for female rulers.