Wonder Boy began as an Arcade title in 1985 by Japanese developer Westone. Hudson later bought the rights to publish the game on the Famicom/NES and renamed it Adventure Island, including creating all-new graphics for the version. The initial Adventure Island was the original Wonder Boy in all but appearance, but future Adventure Island games would become an entirely separate series with no connection to Westone's Wonder Boy. Sega also published Wonder Boy for the Sega Master System, but left it intact.
In 1987 a sequel to Wonder Boy - named Wonder Boy in Monster Land - was released in the Arcades, much like its predecessor. As with Wonder Boy, the game was picked up for home consoles by Sega who published Wonder Boy in Monster Land for the Sega Master System as well as Hudson for a game called Bikkuriman World on the newly released PC Engine/Turbografx-16. As with Adventure Island, Bikkuriman World is more or less identical to Wonder Boy in Monster Land but for its character sprites, in this instance using the license of a popular Japanese collectible sticker series instead. Furthermore, Jaleco acquired the rights and released Saiyuuki World - another graphically-edited but otherwise identical version of Wonder Boy in Monster Land, but this time with Journey to the West characters - on the Famicom in Japan only. Saiyuuki World saw a sequel unaffiliated to Wonder Boy, which in the US was renamed Whomp 'Em.
To make matters more complicated, the Sega Master System version of Wonder Boy in Monster Land was named Super Wonder Boy: Monster World in Japan. This was the origin of the Monster World series.
There are two Wonder Boy IIIs: Wonder Boy III: Monster Lair and Wonder Boy III: The Dragon's Trap (or The Dragon's Curse in some cases). Monster Lair is the "official" Wonder Boy III, as it was released in Arcades first, like its antecedents. Monster Lair was ported to the Genesis by Sega and the PC Engine CD by Hudson, who decided to keep the Wonder Boy label intact this time around. Though this was not the case for the US Turbografx-CD release, which was just named Monster Lair without the Wonder Boy name.
The Dragon's Trap, however, is a Sega Master System original that continued the story of Wonder Boy in Monster Land from where it left off. The game was intended to be called Monster World II in Japan (as the first SMS Wonder Boy in Monster Land had the Monster World subtitle in Japan) but the game was never released there - at that point, Sega had stopped supporting the console in Japan and was giving the Genesis their full attention. However, the game was eventually released in Japan on the Game Gear and retained the Monster World II title.
Meanwhile, Dragon's Trap received a Turbografx-16 release via Hudson as simply Dragon's Curse, once again without the Wonder Boy part of its title. Oddly enough, the game was known as Adventure Island for its Japanese PC Engine CD release, despite having nothing to do with the more famous and now entirely separate Adventure Island series.
The next game in the series was the Genesis game Wonder Boy V: Monster World III, known in the US as Wonder Boy in Monster World. This was the first use of the Monster World title in the US, hence no numeral, but it was the third such game in Japan to have that subtitle, after Monster World I (the SMS version of Wonder Boy in Monster Land) and Monster World II (the Game Gear version of Wonder Boy III: The Dragon's Curse). Presumably because there had been two Wonder Boy IIIs, they skipped forward to V. For whatever reason Hudson released the game as The Dynastic Hero for the PC Engine with slightly different graphics.
Finally, there is Monster World IV, which had until just recently been a Japan-only Sega Mega Drive exclusive, but is now available on Wii Shop and on the Sega Vintage Collection: Monster World compilation for current consoles. The game is no longer affiliated with Wonder Boy, as Monster World IV has a new female protagonist named Asha, but is sometimes called Wonder Boy VI regardless.