To shuffle (Japanese: シャッフル Shaffuru) is to have sufficiently randomized a pile of cards, usually the Main Deck.
Each player must thoroughly randomize (shuffle) their Main Deck where the opponent can see. The opponent must shuffle the Deck further and then cut their deck, then return it to the original owner. No additional randomization may be done to the Deck after this point.
All card searchers involve shuffling the Deck. However, some card effects involve shuffling the Deck specifically, like "Card Shuffle". In rare cases, some card may involve shuffling the cards outside the Main Deck, like "Magical Hats".
Anime and manga
In the manga and second Japanese anime, when Yami Yugi and Arkana Dueled, Arkana used the "shotgun shuffle" method in preparation for the Duel, which Yugi noted as a highly discouraged style as it would bend and damage the player's cards. Naturally, the same applies in the real world. It was Arkana's willingness to use this shuffling method that made Yugi realize that Arkana used the stripper technique, since he had no qualms about damaging his own cards. Later, when Gansley used the effect of Ashingray, he too used the shotgun shuffle, though at the time it was a virtual data deck, so no real cards would be damaged; Yami Marik also used this method during his Duel with Mai Valentine, but is later shown shuffling Yami Yugi's Deck normally.
In Yu-Gi-Oh! 5D's, Duel Disks and Wrist Dealers automatically shuffle the Decks contained within them, in addition to being capable of ejecting specific card from the Deck; these features were maintained for all subsequent series. In Yu-Gi-Oh! ZEXAL, Scorch and Chills disable the auto-shuffle function of their Duel Disks for their Tag-Team Duel with Yuma Tsukumo and Shark to allow them to better coordinate their moves. In Yu-Gi-Oh! ARC-V, removing the Deck from the Duel Disk actually cancels the current Duel. InYu-Gi-Oh! VRAINS, shuffling is no longer animated but instead, the Deck glows and the the word "shuffling" appears on the side where the blade is projected from, though this is only seen in older models of Duel Disks which contain physical cards outside of LINK VRAINS. InYu-Gi-Oh! SEVENS, the Deck is shuffled inside the Duel Disk by being deposited further into the body, after which it resurfaces.