Casus Belli is a new gameplay mechanic introduced in Civilization VI that allows you, after you meet certain prerequisites, to justify the wars you declare and get fewer warmonger penalties ( Grievances in Gathering Storm). The alternative to using a Casus Belli is the Surprise War, which will give you a much larger warmonger/grievances penalty than the Casus Belli, except in the Ancient Era in which there is no warmonger penalty. As the game goes on and you enter new eras, the penalties increase on both Surprise War and the various Casus Belli.
Most Casus Belli are unlocked by researching specific civics, as detailed below. As of the Spring 2018 Update, Casus Belli can be used when declaring a Joint War with another civilization. However, there are diplomatic situations in which Casus Belli cannot be used, such as when a leader protests your positioning of troops on his/her borders, and you decide to answer by declaring war.
Types of Casus Belli
While technically not a Casus Belli (as they are defined as justifying a war and Surprise Wars are not justified), this is the standard (and only) form of war for the early game. There are no requirements for declaring a Surprise War - you can do it immediately, whenever you please. Post-Ancient Era, however, you will receive 50% greater warmonger penalties (150 Grievances) for such an outrage, which makes this the worst possible choice from a diplomatic standpoint. It is also worth noting that this is the only type of war that you can declare on a city-state, since they cannot be denounced.
Persian units under Cyrus gain Movement bonuses for the first 10 turns of a Surprise War, and he incurs less warmonger penalties for declaring them. On the other hand, it is not possible to declare a Surprise War on Canada, nor can the Canadians ever declare surprise wars themselves.
This is the first Casus Belli you can use to justify a war. A Formal War differs from Surprise War not because of any moral justification, but because it comes with a warning: it can only be declared after some diplomatic incidents (such as denouncing or being denounced by another leader at least 5 turns prior, or having a leader break a promise that he or she made to you), and as such comes as no surprise to any party. There are no restrictions on which cities you may capture. If it is the Classical Era or Medieval Era, this is the type of war you will want to use in order to avoid heavy warmonger penalties. The warmonger penalties for this type of war are standard (100 Grievances).
This Casus Belli is unlocked with Diplomatic Service. It is used to declare war on a power that has religiously converted one of your cities, thus sparking religious outrage among your citizens. Given the importance of religion in this game, the other leaders tend to become more understanding of such wars. All warmonger penalties are halved (-50%, or 50 Grievances). If you are playing for a Religious Victory and have competition trying to convert your land, this will be the type of war you are going to be using most often.
This Casus Belli is unlocked with Diplomatic Service. It is used to declare war on a power that has captured a city from one of your friends or allies. Declaring war in such case is seen as justified in any Era; and you will gain no warmonger penalties (0 Grievances) for the declaration of war itself, and as long as you restrict your conquering activities and liberate the captured cities to their original owners. This acts as a retaliatory war of some sort; if you have a strong ally that has been weakened, then you can use this to help them get back on their feet.
Scotland under Robert the Bruce gains major Production bonuses when declaring one of these wars, so look out for any possible opportunity to declare one. Try, however, not to actually succeed; if the enemy is still holding the city at war's end, you can declare another War of Liberation shortly after and benefit from more increased Production.
This Casus Belli is unlocked with Diplomatic Service (Defensive Tactics in Rise and Fall). It is very similar to a Liberation War, but this time it relates to your own cities - as long as you declare war on the power that has captured one (or more) of your cities, and you do not capture any other cities, no warmonger penalties apply (0 Grievances). This can also be used as a form of retaliation, or a way of justifying the reentering of a war that ended previously.
This Casus Belli is unlocked with Diplomatic Service (Defensive Tactics in Rise and Fall). 'Protectorates' in Civilization VI are allied city-states (those you are the Suzerain of), and with this Casus Belli you can declare war on the power that has attacked them. Note that this Casus Belli remains active only while the power and the City-State are at war; if they make Peace, or if the City-State gets conquered, the Casus Belli vanishes. As is the case with Liberation and Reconquest Wars, other leaders understand and admire this kind of loyalty, and no warmonger penalties apply (0 Grievances) for either declaring the war or liberating the city-state. Again, you can use this if another force is becoming too powerful and is beginning to take down your neighbors.
This is one of the more modern Casus Belli, unlocked with Nationalism. It is used to declare war on a power that is two or more technology/civic eras behind you. Other leaders accept such a war as a way to "bring civilization" to an otherwise barbaric and backwards country. All warmonger penalties are halved (-50%, or 50 Grievances). This can be used effectively against empires that have been focusing on a different element, such as their military, and gives power to more technologically advanced civilizations. Domination Victory seekers might find this useful when mopping up the weaker nations around the globe.
War of Territorial Expansion
This Casus Belli is unlocked with Mobilization. It is used to declare war on a power that borders your empire, and in whose lands you wish to expand. You must have at least two cities within 10 tiles of 2 of the opponent's cities for this Casus Belli to apply. Warmonger penalties are reduced by 25% (75 Grievances), since leaders frown upon expansionism, but can understand it somewhat. As the name suggests, this will be used to expand your empire when things start to get crowded, and can also be incredibly useful in domination games.
Chandragupta gains access to this Casus Belli with Military Training instead of Mobilization, and his units gain +2 Movement and +5 Combat Strength in the opening turns of one, which helps to expand the Indian empire across the globe.
Rise and Fall Expansion
With the Rise and Fall expansion, some new Casus Belli have been introduced.
War of Retribution
This is one of the first Casus Belli in the game, unlocked with Early Empire. It is another kind of 'revenge' war, which can be declared on a power that has broken a promise that was made to you within the last 30 turns. During a War of Retribution you have a 50% reduction of standard warmonger penalties for the current era for declaring war (50 Grievances) and capturing cities, but razing cities gives you a 200% penalty. Although an enemy may have broken his promise to you, this does not entitle you to burn their cities to the ground in revenge.
This Casus Belli is unlocked with Ideology and represents the expanded role of this game aspect in modern society. It is used to declare war on a power with a different Tier 3 government: since all Tier 3 governments represent a different Ideology, other leaders see a (somewhat) noble purpose in your declaration and are able to condone it. This Casus Belli grants you a 50% reduction in the warmonger penalties for declaring war (50 Grievances), capturing cities and razing cities. The Ideological war is one of the most suitable for late game conquests, since its only prerequisite is for the target to have a different government, and this is very often the case. Of course, it will be impossible if either you or the target doesn't have a Tier 3 government!
Golden Age War
This Casus Belli is available if the player opts for the 'To Arms!' dedication upon entering a Golden Age. There are no restrictions and all warmongering penalties are reduced by 75% (25 Grievances), which arguably makes it the best Casus Belli for a Domination Victory.
However, the specific dedication is available only after the Industrial Era and requires the player to be in a Golden Age.
A Case for War
Declare a formal war.
Some Wine For Your Soldiers?
Playing as Persia, conquer the original Scythian capital within 10 turns of declaring a surprise war on Scythia
The 25th Dynasty
Playing as Nubia, liberate the original Egyptian capital in a liberation war with the capital's conqueror.