How to Catch Mew
Mew in Generation One
Mew was not actually intended to be one of the Pokémon included in the original Pokémon games. Breaking all protocols, Mew was secretly added into the code at the last minute, as an Easter egg for the developers and as something that could be useful during hypothetical promotional events. It turned out that certain glitches in the game could trigger the appearance of Mew (specifically the Japanese-game-only Magikarp glitch) which let the proverbial pink cat out of the bag, and then everything went wild.
There was outrageous hype about Mew, and everyone wanted to figure out how to catch this elusive and mysterious 151st Pokémon. There were a wide variety of far-fetched rumors circulating on the playground and the early gif-filled Pokémon fansites, but these so called "cheats" were largely just cruel pranks or extreme wishful thinking.
The only way to get your own "official" Mew in the Generation I games was to have participated in one of those promotional events, which did indeed happen. In Japan, there was a contest through CoroCoro magazine in which 151 lucky winners got Mew loaded into their games, and a similar contest was held in Nintendo Power magazine which had 1,000 winners. If you won one of these contests, you had to mail them your cartridge, and they would mail it back to you loaded up with a brand new Mew in your possession.
Later, there were Pokémon tournaments and events where the staff could trade you a Mew, using the in-game trading process with a standard Link Cable and standard games chock full of Mews preloaded by Nintendo. These cartridges were taped into the staff's Game Boys with stickers warning that, if you tried to touch their Mew-filled game, you would face legal action! Mew is serious business! At some of these events, the Mew you were traded would be accompanied with a real-life shiny paper certificate to prove its authenticity. Other events didn't give out the certificate but instead applied a "Mew Inside" sticker to the back of your game cartridge, which I think is incredibly cute.
Other events, rather than trade with a standard game and standard Link Cable, had a bit fancier-looking, less-theft-prone, specialized device known as the "Mew Machine". These machines appear to be some sort of Super Nintendo specifically made to let a Game Boy game be inserted and have a Mew traded to it. When Generation II came out, Nintendo thought that interest in Mew would decline (or they wanted it to?), and these machines were changed to distribute Celebi instead. I think Nintendo was very mistaken, especially given that the name "Mew Machine" apparently stuck.
Nintendo said that apart from these (long over) events, there was no other way to get Mew. You could use a cheating device like a Game Shark, but they claimed a Mew obtained through cheating would not be the same as an "official" Mew. This was total baloney in the Generation I games, but Nintendo really didn't want you to use unlicensed cheating devices. However, once all the Mew-related promotional events were over, non-Japanese players were stuck for awhile with no other way to get a Mew other than using the "forbidden" Game Shark. I desperately wanted a Mew back then, but I had missed the events and wasn't going to purchase a Game Shark on principle. I sought out the silly rumors and even attempted many of them, until I eventually gave up and accepted, however begrudgingly, that Mew was beyond my reach.
However, much later (circa 2003), a glitch was discovered that finally made Mew readily available in all Generation I games! At that point, everyone was so resigned to the idea that Mew was unobtainable that it took some time for people to actually try the glitch and for awareness of it to spread. I know I was in no hurry to test it out when I was first emailed about it. Yet, as was eventually discovered, the trick is indeed real and it does indeed work! Although it provides access to any Pokémon (as well as several garbled glitches), it is known affectionately as the Mew Trick because, let's face it, that's what most people are going to be using it for: catching their very own Mew!
Mew in Generation Two
Mew, as I am sure you expected, cannot be found in the Generation II games, and as I alluded to, the focus at events was on Celebi rather than on Mew. However, you can use the Time Capsule to trade a Mew up from a Generation I game. This can be either an "official" Mew or a Mew obtained through other means.
If you do have an "official" Mew, however, I don't recommend keeping it on a Generation II game for an extended period of time. These games' save files are dependent on the same internal battery that powers the clock (which was inside the game, unlike DS games where the clock is inside the console), so the clock could run out the battery and you could lose your save file, eep! Your Mew is probably much safer stored inside a Generation I game.
Mew in Generation Three
Owning an "official" Mew once again became a hard-to-get prize in Generation III. The glitches that enabled obtaining a Mew in the Generation I games were not carried over to their remakes, FireRed and LeafGreen. Game Freak clearly had to rewrite a lot of their code, so the newer games don't have the plethora of exploitable bugs the first games did, meaning it's not so easy to trick the game into letting you encounter a Mew without actually using a cheating device. Also, trading up is impossible with these games, so you cannot send a Mew from Generation I or II games to the Generation III games, whether that Mew was "official" or not. I can imagine how frustrating it must have been to have an official Mew stuck on an old Game Boy game with no way to send that Mew up to your new Game Boy Advance game.
Mew wasn't officially available in Generation III until Pokémon Festa, held in various locations in Japan during the summer of 2005. At this event, the item called Old Sea Map could be transmitted to Emerald version games and, with this item, you could access the Emerald-exclusive area known as Faraway Island.
On this remote, grassy island, Mew hides in the grass and plays hide-and-seek with you until you manage to corner it and force it into battle (giving you the chance to catch it).
The Old Sea Map was never distributed again and never distributed outside of Japan. My best guess for why is because the Old Sea Map was only good for players with the Emerald version specifically, which means players with other Generation III games would be left out during Mew events. In later events, Mew was distributed via simple trade again, so any Generation III game would work.
However, it seems pretty silly to me that this entire area was coded into the Emerald version, with the cute idea of needing to chase Mew around, and then never really used. What would make me happy would be if you had a rare chance of say, Mr. Briny randomly discovering the Old Sea Map and inviting you to go find the island with him (kinda like a better implementation of Mirage Island, and with an actual reward). That way, Faraway Island could be a perk of owning an Emerald version that anyone could conceivably access, if you keep playing the game for long enough, rather than as just a prize exclusive to some Japanese players who were able to attend an event. I think Mew being a rare secret that anyone could unexpectedly discover is much truer to what caused all the hype about Mew in the first place, rather than it being a prize for going somewhere on a certain day, but I digress...
As I alluded to, there were other events distributing Mew, particularly as promotions relating to the movie Lucario and the Mystery of Mew. For example, to celebrate the US DVD release, Mew was available in official trades at United States Toys 'R' Us stores on September 30, 2006 from noon to 3pm (or until they ran out). Yes, that was for a maximum of three hours and only on one day. I can just imagine being a kid and trying to explain to my mom why we really needed to go to a toy store during that specific time window.
So, yeah, with Nintendo's marketing tactics, I really don't blame people who use cheating devices to get a Mew...
Yet, it should be noted, in this Generation, a "cheated for" Mew might actually be different from an "official" Mew, finally making good on Nintendo's threat way back in Generation I. Specifically in the FireRed, LeafGreen, and Emerald versions, a Mew not met during a "fateful encounter" will disobey, regardless of how many badges the player has. There are workarounds for this problem, but it does make it an extra bit harder to cheat for a Mew when it already was pretty tough to get an "official" Mew in this Generation! I do think that is harsh of Nintendo, although I do like the idea of Mew being impossible to control if it doesn't feel like obeying. It certainly fits Mew's personality to pretend not to notice your orders and just loaf around instead!
Mew in Generation Four
An "official" Mew could be obtained using the WiiWare game My Pokémon Ranch, released in 2008. The game involves transferring Pokémon from your DS to the Wii so you can watch them play on a ranch and interact with your Miis. It only works with the Diamond or Pearl versions (unless you have the expansion that adds Platinum compatibility, which is only available in Japan). Once 999 Pokémon are transferred to the ranch, the ranch owner Hayley will offer to trade you her Mew in exchange for any Pokémon Egg. This Mew will be level 50 and have an unusual moveset. I have not seen the game myself, but it seems really boring and useless, like a glorified screen-saver. The only reason I could see to own it is to get the Mew (and the Phione that is available once 250 Pokémon are transferred), but I figure you might as well get a cheating device at that rate. But, anyway, if you get Mew in My Pokémon Ranch traded to your Diamond or Pearl game (or Platinum game if you have the Japan-only expansion), you can then also trade that Mew to any other game in this generation.
An official event Mew was available via Nintendo Wi-Fi Connection Mystery Gift for North American and PAL versions of the Generation IV games from October 15 to October 30, 2010. Welcome to the future: with this event, Mew could be transfered to your game over the internet, from anywhere with a suitable wireless internet connection during those 15 days. The only thing you might need to bug mom for would be tweaking your router settings or taking you somewhere with free wireless internet access (like the public library). Much better!
Mew can also be transferred to these games from a Generation III game using the Pal Park. Mew will apparently appear in the Forest area.
Note that "cheated for" Mews will only disobey in Generation III. This anti-cheating deterrent was was dropped in all future games.
Mew in Generation Five
Mew is transferable to these games from a Generation IV game using using Poké Transfer (so you can transfer up a Mew obtained in Generation IV or doubly transfer up a Mew obtained in Generation III).
Mew in Generation Six
Mew is transferable to these games from a Generation V game using using PokéTranporter (so you can transfer up a Mew obtained in Generation V or doubly transfer up a Mew obtained in Generation IV or triply transfer up a Mew obtained in Generation III).
I bet there will be some sort of promotion too. We'll see!