After defeating all the monsters in an area (i.e. on a trail), you will be rewarded with a chest containing the design for one epic weapon, plus all the parts needed to build it. Epic weapons have special boosts which are applied to all of your fighters. Your epic weapons appear under "DESIGNS" in the menu.
Epic Weapon Order
In the game menu, Epic weapon designs are listed in folders according to basic weapon type, in the following order:
Many of the epic weapons boost for different elements. At least one weapon in each category boosts Durability. One weapon will boost Luck. At least one weapon boosts for Power. Swords, Pikes, and Axes are an exception for Power. Two Epic Swords, Pikes, and Axes boost for Power.