In order to capture a monster in Asgard Legend, you'll need a monster tamer item. These can be obtained as monster drops or through crafting with Mortenz the Scientist. There are quite a few monsters to capture and the stronger the monster is, the harder it will be to tame. Once a monster is tamed, it will appear in the player's inventory in egg form and require an incubator to hatch.
All pets start off as a level 5 pet with 50 intimacy. In order to raise the pet's level you will need to increase your pet's intimacy with you. This can be achieved by feeding the pet the proper food. Be careful not to overfeed as this will actually decrease your intimacy with the pet. As your intimacy grows, your pets level increases allowing it to deal more damage in battle as well as perform certain abilities at various intimacy levels.
Pet names, intimacy, and levels are all linked to the pet and will retain their values upon being traded. However, pets cannot be more than 2x the player's own base level. This means that opening a level 80 pet on a level 10 novice will result in a pet that's only level 20.
Pets serve two main functions in Asgard Legend: providing additional damage to the enemy and an occasional unique stat boost once loyal. Pets will occasionally help the player attack as long as the player attacks first. The amount of damage dealt to the monster depends on both the type of pet and the pet's level. Each pet also has a unique ability once the pet has reach loyal status. Unique abilities can come in the form of support skills, offensive skills, stat bonuses, and more.
Some pets also have the ability to evolve allowing the player to obtain otherwise impossible to tame monsters. Not all pets can be evolved, however. Once your pet has reached level 90 and has a loyal status you can show Mortenz your pet. Mortenz, located in Yuno, will assess your pet and let you know what needs to be done to evolve it if it can be evolved. Once the pet is evolved, it will start over at level 5 and require proper raising again. Evolved pets will do more damage and provide an improved ability at loyal status.