Legendary, Mythical, Ultra Beast and Paradox Pokémon
The special groups of rare Pokémon: how each is defined, how many there are and how they differ.
Pokémon uses several labels for rare and powerful species, and they are easy to mix up. Each group has its own role in the games and its own rules for obtaining and using it. All counts below are as of Pokémon Legends: Z-A.
Legendary Pokémon
Legendary Pokémon are rare, powerful Pokémon that feature in the myths and stories of their region. There is no official rule that defines them. The label comes from the games, anime and other official media. By Bulbapedia's count there are 71 Legendary Pokémon, and every generation so far has added at least three.
They tend to come in groups: the three birds, the three beasts, the Regis, the lake guardians, the Swords of Justice, the Treasures of Ruin and so on. In the games they are usually single-use encounters, often at the end of the story, and most cannot breed.
Special Pokémon in competitive play
Some legendaries, mostly the cover stars of the games, are called "Special Pokémon". They are restricted or banned in many official tournament formats. A cover legend such as Koraidon or Miraidon is a typical example.
Mythical Pokémon
Mythical Pokémon form a separate group. Traditionally they could only be obtained through special events, and many still need a Mystery Gift. There are 23: Mew, Celebi, Jirachi, Deoxys, Manaphy, Phione, Darkrai, Shaymin, Arceus, Victini, Keldeo, Meloetta, Genesect, Diancie, Hoopa, Volcanion, Magearna, Marshadow, Zeraora, Meltan, Melmetal, Zarude and Pecharunt.
In Japan the two groups were always separate. In the West, they were all called Legendary until Generation V. Not every event Pokémon is Mythical, and not every Mythical is event-only now: several can be found during normal play. Manaphy and Phione are the only members of either group that can breed.
Ultra Beasts
Ultra Beasts come from another dimension, Ultra Space, and arrive in Alola through Ultra Wormholes. There are 11 of them (12 if Necrozma is counted in some media): Nihilego, Buzzwole, Pheromosa, Xurkitree, Celesteela, Kartana and Guzzlord, then Poipole, Naganadel, Stakataka and Blacephalon.
They all share the Ability Beast Boost, which raises their best stat when they knock a Pokémon out. They have lopsided stat spreads and are hard to catch with ordinary balls. Ordinary Poké Balls (the Master Ball excepted) have only a 0.1x catch multiplier on them, while the Beast Ball has 5x. Bulbapedia counts them as a group of their own, outside its 71 Legendary Pokémon.
Paradox Pokémon
Paradox Pokémon are from Scarlet and Violet, where they live in Area Zero. In Scarlet there are ancient relatives of modern Pokémon (Great Tusk, Scream Tail and others, plus Koraidon), and in Violet there are futuristic relatives (Iron Treads, Iron Bundle and others, plus Miraidon). There are 22 in all, 11 of each kind.
They have no gender and cannot breed or evolve. Their base stat totals are 570, 590 or 670, and their Abilities (Protosynthesis for ancient, Quark Drive for future, with Koraidon and Miraidon as the exceptions) are tied to sun or Electric Terrain. Their names are unusual: Great Tusk, Iron Valiant and so on, with Koraidon and Miraidon being the exceptions. Booster Energy can switch their Abilities on without sun or terrain.
A word of caution
Because these labels come from official media and not from a rule, fans sometimes disagree about borderline Pokémon, for example whether Type: Null counts as a sub-legendary. Rely on the game text when in doubt.