A Yu-Gi-Oh! Deck can contain anywhere between 40 and 60 cards, however most professional players play with 40 cards, since they have a greater chance of drawing the card they need each turn, that could win them the game.
Fusion Monsters and Synchro's do not count as your deck, they go in your Extra Deck, as they are now known in 5Ds. If you look on a 5Ds play mat it says Extra Deck, if you look on an original it says Fusion Deck.
Fusion Material Monsters and Tuner Monsters however, DO count as your deck and they must be placed within your main deck.
Ritual Cards must also be in your main deck, as they do not fall into the category that allows you to place them in your Extra Deck, no idea why since they can't be summoned without a Ritual Spell Card.
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The maximum is 60 in the main deck.
15 15
You may have up to three in your main deck and side deck combined.
40 cards
In Yu-Gi-Oh, a sideboard is called a side deck where you put extra cards in to switch out in between duels. However, a side deck must contain 15 cards or 0 cards.