Legendary mechanics in this game is different from original games. There are no legendaries in their static locations and you cannot completely catch some legendaries in this game.

However, there are 5 "King of the hill" legendaries in their own regions, you can catch & own for a temporary time, and you will be forced to accept any PVP battle requests (3v3 battle) to keep the ownership of those legendaries:
- Kanto : Mewtwo (Cerulean Cave)
- Johto : Lugia (Whirl Islands)
- Hoenn : Rayquaza (Sky Pillar)
- Sinnoh : Arceus ( Mt. Coronet)
- Unova : Keldeo ( Moor of Icirrus)
While having one in your party you cannot fly, bike, use the pc or leave the region.
If no one has them at the time then you are able to find them as odds based single encounters in their respective locations.

You can catch Zekrom in Unova storyline and can use it for 2 battles but lose it afterwards and then Zekrom will stay in your room, unused.

About Shaymin, it's exclusive to the Lunar New Year event, you cannot catch it in other ways outside of that event, but you can buy one from Global Trade Link.

Shaymin's IV's can be changed using Gracidea flowers.

Out of legendaries that you can permanently keep, roaming legendaries can be found as odds based single encounters in currently Johto and Kanto regions. The exact odds are unknown but stated by the developers to be between 1/1001 and 1/7999 odds. You cannot sweet scent, fish or repel trick for them. They rotate monthly and you can only catch 1 per month, per region.

Johto : Suicune, Raikou, and Entei, it can be found randomly in GRASS, OUTSIDE AREA (except safari zone) of Johto region after you have beaten the Ho-oh boss battle during storyline. Suicune can also be found in water patches using surf unlike the other 2 Johto beasts.

Kanto: Articuno, Zapdos and Moltres can be found randomly in GRASS/WATER, OUTSIDE AREA (except safari zone) of Kanto region after you obtained 8 badges.

Note: Any area you can use fly or teleport in is an outside area (exception being Ilex forest in johto which is technically a cave but still whitelisted for roaming legendaries).

For Suicune, Raikou and Entei their IV's can be changed using Rainbow quills.
For Articuno, Zapdos and Moltres their IV's can be changed using Elemental Orbs.

Forum guide: https://forums.pokemmo.com/index.php?/topic/160729-roaming-legendaries-faq/

Darkai is a summonable legendary that you can summon temporarily using the "Darkai summoning orb", the orb is obtainable and exclusive only to the Halloween event.

A summonable legendary can be used for a temporary time that extends with each summon, (example: 1 hour long summon and 7 day cooldown) and gains levels growing stronger each time.
They are significantly stronger than roaming legendaries so they are restricted from competitive play and some other game modes.
The summon cooldown can be reset using the Mystic Watch that's an event exclusive item.
Darkrai's IV's can be changed using Dream Essence.

Legendaries/Mythicals cannot breed, but you can alter their IV's with specific items and use Nature herbs to change their nature.

You cannot catch other legendaries currently. However, you can get their "seen" dex from some story lines or later from Gym rebattles / E4 rebattles / Battle Tower battles.