Quests are another integral part of Sword Quest.(2nd part of the title really..)
There are 5 main quests.
The Hero Quests are the main quests. This helps to give a story line to the game as well as gives you a reason and some motivation to improve your skills as it allows you to see the direct impact you have on another because of your rising skill. There is no time limit on your Hero quests.
The Kingdom Quests as it states are randomly picked. It can be any of the coin bought swords from the shop(up to what you have unlocked), and will reset at the start of every new day if you have completed the previous one. There is no time limit on this type of quest.
The Candy(Weekly) Quests are similar to the kingdom quests in that it picks a random material. The biggest difference though is that the weekly quests do have a time limit. You have 1 week. The other difference is that not only does it only require 2 swords from the material it asks for, but it also gives you 1-2 free materials.
The Special Quests is the last real type of one. These quests can be unlocked by completing X number of total swords, or by turning in a high level sword from a specific material. The special quests generally give either gold, hammers, or materials as the reward, and often a mix of the 3. The clues for solving these quests can be found in the Mangas. There is no time limit on your Special quests.
While not a quest, there is a minor "quest" that you can do while Saving Swords. The pink-haired lady who helped you out at the beginning initially challenges, and then encourages you to create more swords. As you do so, she gives you rewards for reaching certain milestones.
The latest Adventurer Quest (Guild) was released on May 9th, 2013. These quests can be unlocked by completing 3 Stars on any of the participating locations with 3 Boss. Your goal will be to vanquish these boss and as a mean to motivate you to advance the adventure mode for better rewards. The added bonus is all of these daily quest gives over 1000 Exp to both you and your pets that'll help players who have no more first kill bonus to level up.