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How to Fusion Summon
A player can Fusion Summon a Fusion Monster from the Extra Deck by activating a card or effect that performs a Fusion Summon. When that effect resolves, the player uses appropriate Fusion Materials listed on a Fusion Monster by moving them from some location(s) to another, as specified by the effect performing the Fusion Summon (if the destination is not specified, then the materials are to be sent to the Graveyard). After that, the player Summons the Fusion Monster from the Extra Deck to any of their unoccupied Monster Zones in face-up Attack or Defense Position.
Even if all of its controller's Monster Zones are occupied, a Fusion Monster can still be Fusion Summoned as long as at least one of the Fusion Materials used is in one of these zones, so that there is an available zone when the Fusion Monster is Summoned.
List of cards that Fusion Summon
Example A: Normal method
Example B: Using Fusion Substitute Monsters
+ a correct Fusion Material
("Elemental HERO Sparkman")
Example C: Using a monster as both a Fusion Material and to conduct a Fusion Summon
A Fusion Material with an effect that Fusion Summons
("D/D Swirl Slime")
+ A Fusion Material
Like other Extra Deck monsters, if a Fusion Monster is not Fusion Summoned first, it cannot be Special Summoned from the Graveyard or while banished (unless it specifies another way to properly Special Summon it, in which case it must first be Summoned that way).
For example, if a Fusion Monster is Special Summoned from the Extra Deck by the effect of "Cyber-Stein" and sent to the Graveyard, or was sent to the Graveyard by "Fusion Guard", it cannot be Special Summoned from the Graveyard by "Re-Fusion"; however, if it was originally Fusion Summoned by "Polymerization", it can be Special Summoned from the Graveyard by "Re-Fusion".
In the anime
While visual depictions of Fusion Summons have varied significantly among the first four anime series, most generally show the Fusion Materials, or their cards, being warped into particles or forms of energy in a spiral direction, which then merge with each other within a spiraling portal, from which the Fusion Monster emerges. One depiction, notably used for "Dark Paladin" and "Dragon Knight Draco-Equiste", shows the Fusion Materials being engulfed in a sphere of energy, which then disperses to reveal the Fusion Monster. Other times, the Fusion Monster emerges from a fierce whirlwind.
It was not until Yu-Gi-Oh! ARC-V that a standardized animation is adopted for all Fusion Summons: a blue and orange rotating background based on the artwork of "Polymerization", and the Fusion Materials transforming into featureless shapes of different colors (usually contrasting ones like blue and red, yellow and violet, etc.) that swirl together to form a spinning spiral, which then sometimes turns into a bright white light. It is then followed by a close-up shot of the player, usually clasping or clapping their hands together, as if mirroring the materials becoming one.
In Yu-Gi-Oh! VRAINS, Fusion Summoning is changed to a different CGI effect, where the Fusion Materials transform into streams of colored particles that swirl together and then burst upwards into vortexes of colors matching the streams, which twist around each another and quickly merge into a large purple one, from where the Fusion Monster appears.
In the Yu-Gi-Oh! anime, under Battle City Rules, which applied during the Battle City tournament, Fusion Monsters cannot attack the same turn they are Summoned. This is exemplified in episode 65, when Strings uses "Quick Attack" on "Humanoid Worm Drake" to bypass that rule in his Duel against Yami Yugi. This rule does not exist in the OCG/TCG and was removed for all subsequent series.
In Yu-Gi-Oh! ARC-V, a Fusion Summon performed using Fusion Materials that have been previously Pendulum Summoned is referred to as a "Pendulum-Fusion".