As the name Mystery Dungeons implies, dungeons are the prime focus of this two pokemon games. One striking feature about dungeons is that the same location changes everytime you enter, meaning the layout, item distribution, and many other aspects will reset with every new access. However, the total number of floors, the type of items available, as well as pokemon species that appear in them will remain constant.
You are first introduced to the dungeons in the game when you and your partner agreed to help a Butterfree find its Caterpie child who fell into the deep cracks of the ground, and you should take time to familiarize yourself with how different things work in this introductory location, known as the Small Forest.
The basic thing to do is to walk around until you find the stairs to go to another floor (you can walk diagonally easier by holding 'R'). There is an outline of the layout of the floor you're on overlapping the active image, where your leading pokemon is a white spot, your other team members yellow, the enemies red, items or stat restorers blue, and most importantly, the staircase as a blue square. You can press 'select' to get a clearer view of this outline picture.

There will be items scattered on the ground which you can pick up, and wild pokemon will attack you. The game adopts a turn-based mechanics like in previous games, which means your enemies will only move when you make a move (such as walk a step, do an attack, or use an item).
There are several ways to attack in a dungeon. The most direct method is to press 'A', which deals the least amount of damage in the form of a tackle. This kind of attack has no PP limits, but note that it does not harm Shedinja.

The second way to attack is to actually use your pokemon's known attacks, either accessing it from the menu or short-keying one of them and unleash by pressing 'L' and 'A' (Check Main Menu section for details).
The third way is to use throw-items that you have in your bag, such as stones and spikes (these are able to hurt Shedinja). Again, you could directly access them from the menu or shortkey them.
Remember that the direction you are facing while doing any attack is important, as your character may not face the foe you wish to attack sometimes. Unless you're wedged between the wall and your enemy or something, do not change your directions with the direction key pad, but rather, press 'start', then use the direction key pad, and you'll be able to face the direction you wish.
Your HP will heal itself as you walk around, but your hunger index will drop. Keep yourself full by eating food you picked up in the dungeons, because once your hunger index becomes too low, you will lose HP while you walk around instead.
For the first half of the game (before the credits roll), your partner will follow you everywhere you go, and you must keep your partner and yourself alive in every mission. You cannot control what your partner does, but from its attack list, you could press 'select' on the moves that you allow it use only (a star will appear next to it), so it won't pull off pointless attacks such as tail whip and growl all the time.

During a battle, if you've been inflicted with poison or burn, the effect will remain until you cure yourself with an item or go to another floor. Ones like confuse, attract and paralyze will resolve with several steps. If you've been hit by a stat lowering attack (such as tail whip), there will be a yellow arrow on top of your pokemon. Entering another floor will reset that stat, or you can walk over one of those big green arrows to cure yourself of that too.
As you progress to the later dungeons, there will be special areas in the dungeons which are accessible only to certain pokemon. One, the lava ground, which only fire type pokemon can walk on.
Two, water, which only water pokemon can travel on. Sometimes there are hidden items in these areas, so you could try to explore them. Later in the game, you may also need such abilities to get across water barriers which surround certain key items. The rest of your team will remain along the edge of these areas, and may wander off to other parts of the dungeon. When you walk back to your partner or team members at the edge, the game will automatically prompt you to temporarily send them to another part of the dungeon to wait. However, this way, it will be quite hard to monitor how your other members are doing, so avoid doing this.
Note that some flyers, levitators and floaters are able to travel on both lava and water, though there is no strict rule on this. Basically, just look at the animation of that pokemon when it moves. If it can lift itself completely off the ground, chances are it can, though it's not always the case. However, remember that non-fire type pokemon will get burnt if they travel over lava ground.
Some ghost pokemon are special in the sense that they can even walk through walls (with the exception of more solid looking ones such as Sableye and Dusclops). Other pokemon can do this when equipped with the special Passthrough Scarf. Yet, your hunger index drops nearly 5 times quicker when you travel in walls.

For the later dungeons, there will sometimes also be requirements or limitations to access them. For instance, you will need certain HMs or pokemon that know the particular attack to enter certain locations, and some dungeons limit the total amount of team members you can bring along with you on that mission. A message will show up to refuse your entry to that dungeon if you don't fulfill the requirements.
Most of the dungeons are involved in the main plot of the game, and are unlocked one after another. The difficulty and number of floors will increase as you go along, and there will even be traps that induce various effects in those dungeons that come after the credits.
Finally, remember you cannot stay on any single floor in the dungeons for too long. There is a fixed maximum number of steps that you can take in each floor, the when the number hits zero, your current mission will be considered failed, and you will be teleported out of the dungeon. There are warning signs when this is about to happen, a total of three of them. When a message suddenly appears out of nowhere, and you see leaves passing by, you better locate that staircase and leave as soon as possible. When the leaves blow by the third time, your time is up

 
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