There's a world map, but it's a menu. It's like a slightly more interactive / differently presented Puzzle and Dragons. You select the region you're going into, then select the part of that region you wish to go to.
Dungeon gameplay is like a 2D version of the modular room system of Dragon Quest Monsters: Terry's Wonderland. Dungeons are set up in "rooms" that have N / S / E / W exits. When enter a room (you move by swiping the screen in the direction you want to go) you will see pots, chests, glowing things on the ground, helpful monsters and signs, or enemies. Enemies attack automatically, the rest require you click on them.
The battle engine is a ported version of the standard DQ engine. There's an AI you can enable, or you can do standard combat.
It appears that like Puzzle and Dragons monsters have an active skill (and a host of magic spells - for example, my Slime Knight can cast heal) and Leader skills which activate when they are party leader. My Slime Knight causes Slimes in the party to get 5% extra HP, for example.
The game apes the Puzzle and Dragons stones and Pal Points systems, although they scale the numbers for stones by 100 - you spend 500 stones for a magic map, or 200 pal points for a lesser one. The Magic Maps are like the egg machine lotto from P&D except they put you in a special four room dungeon (entrance, empty room, boss, exit) that requires you fight the monster on the map to add them to your party. No clue if you can retry.
I highly suspect that empty room has a chance to spawn stuff like Metal Slimes or something.
