Each player (including you) reveals the top 3 cards of their deck, sets one of them aside that you choose on their Dungeon mat, and puts the rest back on top in an order they choose.
After a player shuffles, they may put any card from their mat into their discard pile.
(All return to their deck at the end of the game.)
When you gain this, each player takes a Dungeon mat. This card affects all players, including you.
Each player shuffles their discard into their deck if necessary to get 3 cards to reveal, or just reveals as many as they can if they still have fewer than 3 cards. You pick one of the revealed cards from each player and they must place that card on their Dungeon mat, face up. The cards remain face up on their mat, and can be viewed by anyone. They do not have to be kept in any particular order. The remaining revealed cards are put back on top of their deck in any order the player wants, without showing the order to the other players.
There is only one way a player can move a card from their Dungeon mat. When they form a new deck because they have run out of cards, after they shuffle and place the cards into their new deck, they may choose any one card from their Dungeon mat and place it in their discard pile. This means that card will not cycle into that player's deck until they have played through their newly formed deck.
In cases when a player does not have any cards in their discard pile with which to form a new deck, they are still allowed to move a card from their Dungeon mat to their discard pile. The event which triggered the formation of the empty deck will still fail to find a card, but the next such event will get that card which was released from the Dungeon mat.
For example, if a player has no cards in her deck or discard piles and she plays a Laboratory: she attempts to draw a card and would have to form a new deck, which would still be empty, and so she would draw nothing. At that point, she could place a card from her Dungeon mat into her discard pile. Then the second draw would happen, and she shuffles that one card from her discard pile into her deck and draws it. Once again, another card could be released from the Dungeon, although it would not be drawn by this Laboratory, because it had completed resolving.
Only a maximum of one card can be put into the discard pile when the opportunity arises, but there is no obligation to do so. Cards can stay on the Dungeon mat until the end of the game, when they return to their player's decks for scoring.